RSG CLASSES

ECONOMICS (H) SEM-1

INTRODUCTORY STATISTICS OF
ECONOMICS

BY RAHUL SIR
(SRCC GRADUATE, DSE ALUMNI)
1 - 13

14 - 18

19 - 42

43 - 65

66 - 88

88 - 99
1|Page RAHUL STUDY GROUP

CHAPTER-1
MEASURE OF CENTRAL TENDENCY
Arithmetic Mean
Individual Series
1) On a particular warm day, six shops reported early morning sales of
cold drinks as follows:
Shop: A B C D E F
Sales: 73 80 36 75 68 82

Calculate the appropriate of central tendency and explain the reason for
year choice.
[DU B.A. Eco (H), 2010 Marks] [Ans.: A.M. = 69]
2) Find the arithmetic mean of the numbers 1,2,3………………….., n.
𝐧+𝟏
[𝐀𝐧𝐬. : ]
𝟐
Discrete Series
3) Find the arithmetic mean from the following frequency table:
Marks 52 58 60 65 68 70 75

No of 7 5 4 6 3 3 2
Students

[Ans.: 61.6]
Continuous Series
4) The data on number of patient attending a hospital in a month are given
below. Find the average number of patients attending the hospital in a
day, using direct method
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No. of patients 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60

No. of days 2 6 9 7 4 2

Attending the hospital

[Ans.: 28.67]
5) In a study on patients, the following data were obtained. Find the
arithmetic mean, using step-deviation method
Age (in 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89
years)

No of cases 1 0 1 10 17 38 9 3

[Ans.: 60.7]

6) The sum of deviations of a certain number of observations measured

from 4 is 72 and the sum of the deviations of the same values from 7 is -
3. Find the number of observation and their mean.
[Ans.: 25 and 6.88]
7) The sum of the deviations of a set of value X1, X2, ……, Xn measured from
50 is -10 and the sum of the deviations of value from 46 is 70. Find the
value of n and the mean. [Ans.: n = 20 & 𝑿 ̅ = 49.5]
Finding the Missing Items
8) Find the value of p for the following distribution whose mean is 16.6.
X: 8 12 15 P 20 25 30

f: 12 16 20 24 16 8 4

[Ans.: p = 18]

9) The mean of the following frequency distribution is 50. But the

frequency f1 and f2 in classes 20 – 40 and 60 – 80 are missing. Find the
missing frequency.
Class 0-20 20-40 40-60 60-80 80-100 Total

Frequency 17 f1 32 f2 19 120
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[Ans.: f1 = 28, & f2 = 24]

Application based question

10) The mean age of 20 students of a class is 12 years. A new student is
admitted to that class and the mean goes up, to 12.5 years. Find the age
of new student. [Ans.: 22.5 years]

11) The mean of seven numbers is 10; the mean of the first four numbers is
8 and that of the last four numbers is 16. Find the 4th number.
[Ans.: 26]

12) Average marks in Statistics of 10 students of a class was 65. A new

students took admission with 20 marks whereas two existing students
left the college. If the marks of these students were 40 and 39, find the
average marks of the remaining students. [Ans.: 65.67]

13) The daily average sales of a store were Rs. 2,750 for the month of Feb.
1996. During the month, the highest and the lowest sales were Rs.
8,950 and Rs. 580 respectively. Find the average daily sales if the
highest and the lowest sales are not taken into account.
[Ans.: Rs. 2600.74]
14) The mean marks obtained by 300 students in the subject of Statistics
are 45. The mean of the top 100 of them was found to be 70 and the
mean of the last 100 was known to b 20. What is the mean of the
remaining 100 students? [Ans.:45]
-
15) A firm of readymade garments makes both men’s and women’s shirts.
Its profit averages 6% of sales. Its profits in men’s shirts average 8% of
sales: and women’s shirts comprise 60% of output. What is the average
profit per sales rupees in women’s shirts? [Ans.:4.67%]

16) B.Com. (Pass) III year has three Sections A, B and C with 50, 40, 60
students respectively the mean marks for the three sections were
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determined as 85, 60 and 65 respectively. However, marks of a student

of section A were wrongly recorded as 50 instead of zero. Determine
the mean marks of all the three sections put together. [Ans.: 70]

17) The mean weight of 150 students (boys and girls) in a class is 60 Kg.
the mean weight of boy students is 70 Kg and that of girl students is 55
Kg. find the number of boys and girls in that class.
[Ans.: Boys: 50 & girls: 100]

18) The average monthly wage of all workers in a factory is Rs. 444. If the
average wages paid to male and female workers are Rs. 480 and Rs. 360
respectively. Find the percentage of males and females employed by the
factory. [Ans.:70% males, 30% females]

19) In a certain examination, the average grade of all students in class A is

68.4 and that of students in class B is 71.2. if the average of both classes
combined is 70, find the ratio of the number of students in class A to the
number in class B. [Ans.3:4]

Property 1: Sum of deviations from actual arithmetic mean is Zero

Property 2: Sum of square of deviations from arithmetic mean is least
Property 3: Change of origin & scale

20) What is the effect on Arithmetic Mean, if

(a) 5 is added to all the values.
(b) 5 is subtracted from all the values.
(c) All the values are multiplied by 5.
(d) All the values are divided by 5.
Explain with the help of an example.
[Ans.: Mean will get: (a) increased by 5; (b) reduced by 5 (c) Multiplied by
5; (d) divided by 5]
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21) The mean of 68 numbers is 18. If each number is divided by 6, find the
new mean. Find the relation between the new mean and the old mean.
𝟏
[Ans.: Mean = 3, and new mean = of old mean]
𝟔

Median: A Positional Average

22) The numbers of runs score by 11 players of the cricket team of a school
are: 5, 19, 42,11, 50, 30, 21, 0, 52, 36, 27
Find the median scores. [Ans.: 27]

23) The numbers of runs scored by 12 players of the cricket team of a

school are: 5, 19, 42, 11, 50, 30, 21,0, 52, 36, 27,60 [Ans.: 28.5]

24) The median of the observation 8, 11, 13, 15, x+3, 30, 35, 40, 43,
arranged in ascending order is 22, find x. [Ans.: 19]

25) Calculate median for the following data:

No. of students: 6 4 16 7 8 2
Marks: 20 9 25 50 40 80
[Ans.: 25]

26) Find the median of the following frequency distribution:

X: 5 7 9 12 14 17 19 21
F: 6 5 3 6 5 3 2 4
[Ans.: 12]

27) The following table gives the weekly expenditure of 100 families. Find
the median weekly expenditure;
Weekly Number
of
Expenditure
families

0-10 14

10-20 23
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20-30 27

30-40 21

40-50 15

[Ans.: 24.81]
28) The following table gives the marks obtained by 50 students in
Economics. Find the median.
Marks No of student Marks No of students

10 - 14 4 30 - 34 7
15 - 19 6 35 - 39 3
20 - 24 10 40 - 44 9
25 - 29 5 45 - 49 6

[Ans.: 29.5]

29) Compute median from the following Data:

Mid- values: 115 125 135 145 155 165 175 185 195
Frequency: 6 25 48 72 116 60 38 22 3
[Ans.: 153.79]

30) Compute the appropriate average for the following data giving reason
for your choice
31) :
Values: 0-100 100-200 200-300 300-400 400 & above
Frequency: 40 89 148 64 39
[Ans.: 241.22]

32) An incomplete distribution is given below:

Variable: 0-10 10-2020-3030-4040-5050-6060-70
Frequency: 10 20 ? 40 ? 25 15
(i) You are given that the median value is 35. Find out missing frequency, if
total frequency = 170.
(ii) Calculate the arithmetic mean of the completed table.
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[Ans.: (i) f1 = 35, & f2 = 25, (ii) 35.88]

Partition Values
33) Compute Q1, Q2 and Q3 of the following data:
9, 13, 14, 7, 12, 17, 8, 10, 6, 15, 18, 21, 20
[Ans.: Q1 = 8.5, Q2 = 13 and Q3 = 17.5]

34) Compute D3 of the following data:

13, 14, 7, 12, 9, 17, 8, 10, 6, 15, 18, 21, 20
[Ans.: D3 = 9.2]

35) Compute D3, D6 and D7 of the following data:

Marks 10 20 30 40 50 80
No. of students 4 7 15 18 7 2
[Ans.: D3 = 30, D6 = 40 and D7 = 40]

Commute Q3 and D7 for the following frequency distribution:

Marks: 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
No. of student: 3 10 17 7 6 4 2 1
[Ans.: Q3 = 40.83, & D7= = 37.14]

36) Given below are the marks obtained by 50 students appearing for an
admission test:
Marks 0-10 10-20 20-30 30-40 40-50

No. of 6 8 20 9 7
students:

If the cut-off point was 34, find the percentage of students scoring more
than 34 marks.
[Ans.: 24.8%]

37) Revenue from daily sales of 250 shops selling fireworks are given
below:
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Revenue (Rs.) No. of

shops

Less than 500 50

500-1500 120

1500-2500 50

2500-3500 30

If a pollution tax is to be levied on shops earning at least Rs. 1,800 per day,
calculate the % of shops which will have to pay the tax.
[DU B. A. Eco (H), 2010; 2 marks] [Ans.: 26%]

38) Following is the distribution of marks obtained by 100 students

appearing for an admission test:
Marks more than: 0 10 20 30 40 50 60 70
No. of students: 100 92 77 57 40 25 15 5
If 70% of the students passed the test, find the passing marks of the test.
(Hint: find the 30th percentile.) [Ans.: 23.5]
Mode
Case-I: Maximum frequency is the modal frequency.
Case-II: Maximum frequency is not the modal frequency.
Case-I: Maximum frequency is the modal frequency.
39) Calculate mode for the following distribution:
Production per day in tonnes 21-22 23-24 25-26 27-28 29-30
No. of days 7 13 22 10 8
[Ans.: 25.36 tonnes]
Case-I: Maximum frequency is not the modal frequency.
40) Calculate mode for the following distribution:
Variable (X) 10 12 14 16 18 20 22 24
Frequency (f) 4 6 20 32 33 17 6 2
[Ans.: 16]
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41) Calculate mode for the following frequency distribution:

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Frequency (f) 4 6 20 32 33 17 6 2
[Ans.: 40.9]

42) Find the mode from the following table:

Class: 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Frequency: 8 14 25 18 25 10 5
[Ans.: 35]
Multi-Modal Distribution
43) Calculate mode for the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Frequency (f) 4 6 20 32 33 17 8 2
[Ans.: 40.04]
Unequal Class Intervals
44) Calculate mode for the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 50-60 60-80
Frequency (f) 4 6 20 32 33 17
[Ans.: 49.5]

HISTOGRAM
45) Construct a histogram from the following data and out the mode:
Marks: 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40

Frequency: 328 350 720 664 598 524 378 244

46) Prepare a histogram from the following data and find out the mode:
Marks: 0-10 10-20 20-40 40-60 60-70 70-100 100-120 120-130
No. of Student 4 6 14 16 14 8 16 5

47) The following data shows the distribution of marks of 50 students:

Marks No. of students
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0<2 6

2<4 10

4<8 12

8 < 12 14

12 < 14 6

14 < 16 2

(i) Construct a histogram.

(ii) Is the histogram symmetrical?

(iii) What proportion of students get less than 12 marks?

TRIMMED MEAN

48) Find the trimmed 20% mean for the following test scores: 60, 81, 83,
91, 99 [Ans. 85]

49) Find the 10% trimmed mean for the following data
6.5 ,12 ,14.9, 10.0, 10.7 ,7.9 ,21.9 ,12.5 , 14.5 ,9.2 [Ans 11.46]

MISSLENOUS QUESTION
50) The following data consists of information on number of trees in ten
different rows of a reserved forest:

{68, 16, 35, 42, 6, 105, 44, 54, 33, 80}

(i) Find the arithmetic mean and median of the number of trees in
a row.

(ii) If the 6th observation was 95 instead of 105, how would the
mean and median change?
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(iii) Calculate a 15% trimmed mean. [Ans.(i)48.3 (ii)mean 47.3 and

median is same (iii) Trimmed mean 46.25)

51) The cropping intensity on 70 farms in Haryana was found to be as

follows -
Cropping intensity (%) Number of farms

70 - < 100 5

100 - < 120 8

120 - < 140 24

140 - < 150 13

150 - < 170 15

170 - < 200 5

(i) Draw a histogram. Is it bimodal?

(ii) Comment on its shape, (Ans; close to symmetrical)

(iii) What % of farms have a cropping intensity of less than 120%?

(Ans;18.57%)

(iv) What proportion of farms have a cropping intensity of 100% or

more but less than 170% . (Ans; 85.71%)

52) Consider the following sample:

27, 6, 24, 30, 50, 2, 150, 44, 80, 36

Find the arithmetic mean and median( Ans; 44.9, 33)

(i) Find the 15% trimmed mean( Ans; 35.14)

12 | P a g e RAHUL STUDY GROUP

(ii) Which of these three measures best summarizes these data?

(Ans: MEAN MED MODE)

(iii) By how much can the smallest sample observation be increased

so that the sample median remains the same? (Ans: By 28)

(iv) By how much can the largest sample observation be decrease so

that the sample median remains the same? (Ans; By 114)

53) Show that the weighted A.M of first n natural numbers whose weight
𝟐𝒏+𝟏
are equal to corresponding number is equal to .
𝟑

54) The marks of 20 students in a 50 marks mathematics test are given

below:

18 20 25 28 30 35 36 38 39 40 41 41 42 42
43 44 45 45 47 50

Calculate a 10% trimmed mean for the data above. Also calculate the
median. It was later discovered that the student whose marks were
recorded as 35, actually had 45 marks. How will this affect the median
value? [ Ans : 38.375 ,40.5, New median=41]
55) Descriptive Statistics of a data set are given as follows:
Mean=535, Median=500, mode=500, SD=96, minimum=22,
maximum=925, 5th percentile=400, 10th percentile=430,
90th percentile=640, 95th percentile =640, 95thpercentile =720.
What can you conclude about the skewness of the histogram?

56) Difference between descriptive and inferential statistics. Identify which

of the following statement is inferential in nature.
a) In a random sample of 300 people in Delhi, 240 read at least on
newspaper daily
A) 80% of people sampled read at least one newspaper daily.
B) 80% of all the people in Delhi read at least one newspaper in daily
-
b) In a random sample of 100 students in DU are from outside delhi.
13 | P a g e RAHUL STUDY GROUP

A) 60% of students in DU are from outside Delhi

B) 60% of sampled students of DU were from outside Delhi.

57) The salary of 10 employees in a company are given below.

590, 815, 575, 608, 350, 1285, 408, 540, 555, 679

a. Calculate the sample mean and median. [Ans 640.5, 582.5]

b. Suppose the 6th observation had been 985 rather than 1285.
How would the mean and median change?
[Ans mean is 610.5, median will be same]
c. Calculate a 20% trimmed mean by first trimming the two smallest
and two largest observations. [Ans 591.17]
d. Calculate 15% trimmed mean. [Ans 593.705]

58) Calculate the trimmed mean for the below data by deleting the smallest
and largest observation. What is corresponding trimmed percentage?
6, 5, 11, 33, 4, 5, 80, 18, 35, 17, 23 [Ans 17, 9.09%]

59) Blood pressure values are often reported to the nearest 5 mmhg
(100, 105, 110 etc). Suppose the actual blood pressure values for nine
randomly selected individual are

118.6, 127.4, 138.4, 130, 113.7, 122, 108.3, 131.5, 133.2

a. What is the median of reported blood pressure values. [Ans 125]

b. Suppose the blood pressure of second individual is 127.6 rather than

127.4. how does this affect the median value. [Ans median value is 130.that
mean median is highly sensitive to small change]

60) If the mean of a set of observations 𝑥1 , 𝑥2 , … , 𝑥2 𝑖𝑠 𝑋̅, then the mean of

the observation 𝑥𝑖 + 2𝑖; 𝑖 = 1, 2, … , 𝑛 is [Ans 𝑿̅ + (𝒏 + 𝟏)]

61) The weighted means of 1st n natural numbers whose weights are equal
𝟑𝒏(𝒏+𝟏)
to the squares of corresponding numbers is [Ans ]
𝟐(𝟐𝒏+𝟏)
14 | P a g e RAHUL STUDY GROUP

CHAPTER -2

MEASURE OF DISPERSION

Range
1) Find the range and the coefficient of range for the following
observations.
65, 70, 82, 59, 81, 76, 57, 60, 55, and 50 [Ans.: 32, 0.24]
2) The following table gives the age distribution of a group of 50
individuals.
Age (in 16- 21- 26- 31-
years) 20 25 30 36
No of 10 15 17 8
Persons
Calculate range and the coefficient of range. [Ans.: 20 years, 0.385]
Population Standard Deviation
3) Calculate standard deviation of the following marks obtained by 5
students in a tutorial group: 8, 12, 13, 15, 22 [Ans.: 4.6]

4) Find the S.D. of the following frequency distribution:

Class- 10 20 30 40
Mark
Frequency 2 3 5 2
15 | P a g e RAHUL STUDY GROUP

[Ans.: 9.55]
5) Find the standard deviation of the following data using:
(a) Direct method.
(b) Assume mean Method.
(c) Deviation From Arithmetic mean
Class 0- 10- 20- 30- 40-
10 20 30 40 50
Frequency 5 10 15 10 5
[Ans.: 11.54]
Coefficient of variation (the relative Measure of S.D)
6) Find the coefficient of variation, if the sum of squares of the deviations
of 10 observations taken from the mean 50 is 250. [Ans.: 10%]

7) If the mean is 10 and the coefficient of variation 5, then what is the

standard deviation? [Ans.: 0.5]

8) To analysis of the monthly wages paid to the workers in two firms A

and B, belonging to the same industry, gave the following results:
Firm Firm
A B
No of workers 160 150
Average wage 560 575
Variance of wage 400 625
distribution
Find out:
(i) Which firm pays larger amount as monthly wages?
(ii) In which firm is there greater variability in individual wages?

9) A batsman Mr. A is more consistent in his last 10 innings as compared

to another batsman Mr. B. Therefore, Mr. A. is also a higher run getter".
Comment. [Ans.: The given statement is not necessarily true]
16 | P a g e RAHUL STUDY GROUP

10) "After settlement, the average weekly wage in a factory had increased
from Rs 800 to Rs 1200 and the standard deviation had increased from
Rs 100 to Rs 150. After settlement, the wage has become highland more
uniform/' Comment. [Ans.; The wage has increased, but not more
uniform]

11) The arithmetic mean and standard deviation of two Brands of bulbs are
given below:
Brand I Brand II
Arithmetic Mean 800 Hours 770 Hours
Standard Deviation 100 Hours 60 Hours
Calculate the coefficient of variation for the two brands. [Ans.: 12.5%,
7.79%]
12) Show that the coefficient of variation for the first N natural numbers is
𝑛+1
given by √
3(𝑛−1)

Correction the Standard Deviation and Variance

13) Mean and coefficient of standard deviation of l00 items are found by a
student as 50 and 0.1. If at the time of calculations two items are:
wrongly taken as 40 and 50 instead of 60 and 30, find the correct mean
and standard deviation. [Ans.: 50, 5.39]

14) Define Dispersion. The Mean and the Standard Deviation of a series of
100 items were, found to be 60 and 10 respectively. While calculating,
two items were wrongly taken as 5 and 45 instead of 30 and 20.
Calculate corrected variance and corrected coefficient of variation.
[Ans.: 92.5, 16.03%]
Properties of Standard Deviation

15) The following are the various properties of standard deviation:

Property 1: Sum of the squares of deviations.
17 | P a g e RAHUL STUDY GROUP

The sum of the squares of deviations of a set of observations from their

actual arithmetic mean is the least.
16) The sum of squares of deviations measured from mean is the.............
[DU B.Com. (H), 2006 (Corr.)][Ans.: Least]
Property 2: Standard Deviation of 1st n Natural Numbers
17) State and prove the formula for computing standard deviation of first V
𝒏𝟐 −𝟏
natural numbers 1, 2, ..., n. [𝑨𝒏𝒔.: √ ]
𝟏𝟐

18) Find the standard deviation of 1st 10 natural numbers. [Ans.; 2.87]
Property 3: Effect of change of origin and scale.
19) What is the effect on standard-deviation, if:
(a) All the items are increased by 5.
(b) All the items are decreased by 5.
(c) All the items are multiplied by 5.
(d) All the items are divided by 5.

20) If 20 is subtracted from every observation in a data set, then the

coefficient of variation of the resulting data set is 20%. If 40 is added to
every observation of the same data set, then the coefficient of variation
of the resulting set of data is 10%. Find the mean and standard
deviation of the original set of data. [Ans.: 80 and 12]
Property 4: Combined Standard Deviation
21) For a group of 100 items, mean and standard deviation are 8 and √10.5
respectively. For 50 observations selected from these 100 observations,
mean and standard deviation are 10 and 2 respectively. Find mean and
standard deviation of remaining 50 observations. [Ans.: 6, 3]

22) Find the missing information from the following:

Group Group Group Combined
I II III
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Number 50 ? 90 200
Standard 6 7 ? 7.746
Deviation
Mean 113 ? 115 116
[Ans.: 60, 120, 8]

Sample Standard deviation

23) For the Data 29.5 , 49.3 ,30.6 ,28.2 ,26.3 ,33.9 ,29.4 , 23.5 ,31.6. Compute
the following.
a. The Sample Range [Ans 25.8]
b. Sample Variance and Standard Deviation [Ans 49.3112 , 7.0222]

̅ = 76831 s=180 ,
24) Consider the Following information with n=4 . ×
Smallest x=76683 ,Largest x=77048..Determine the Value of Two
Middle observation. [Ans 76683 ,76910]
25) Suppose for 15 observation Sample means is 12.58 and Sample
standard deviation is .512mm. 16th observation is 11.8. Using the Q29
find the Sample mean and standard deviation of all 16 observation
[Ans., 12.53, .532]
̅ n and S2n denote the sample mean and the variance for the sample
26) If ×
̅ n+1 and S2n+1 denote these when an additional
X1, X2 …. Xn and ×
observation Xn+1 is added. Prove that
𝑛
nS2n+1 = (n – 1) S2n + ̅ n) 2
(Xn+1 – ×
𝑛+1
19 | P a g e RAHUL STUDY GROUP

CHAPTER -3

PROBABILITY
Basic Probability

1) Three dice are thrown simultaneously. What is the probability that all
of them show the same face? [Ans.; 1/36]

2) What is the chance that a non-leap year selected at random will have
a. 53 Sundays,
b. 52 Sundays. [Ans.: (i) l/7, (ii) 6/7]

3) The odds in favour of an event are 3:5. Find the probability of

occurrence of this event. [Ans.: 3/8]

4) A card is drawn from a well shuffled deck of 52 cards. What are the
odds in favour of getting a face card? [Ans 3:10]

5) A and B are two events, not mutually exclusive, connected with a

1 2 1
random experiment E. If P(A) = , P(B) = and P(A ∪ B) = , find the
4 5 2
values of the following probabilities:
a. P(A ∩ B)
b. P(𝐴 ∩ 𝐵𝑐 )
c. P(𝐴𝑐 ∪ 𝐵𝑐 ) where c stands for the complement.
[Ans.: (a) 3/20, (b) 1/10, (c) 17/20]
20 | P a g e RAHUL STUDY GROUP

6) Prove that for every two events A and B , show that the Probability that
exactly one of the two events will occur is given by the expression.
P(A) + P(B)- 2P(A∩B)

7) If a card is drawn at random from a pack of playing cards, what is the

probability of getting a king or a red card? [Ans.: 7/13]

8) The probabilities of occurrence of two events E and F are 0.25 and 0.50
respectively. The probability of their simultaneous occurrence is 0.14.
Find the probability that neither E occurs nor F occur. [Ans.: 0.39]

9) A card is drawn from an ordinary pack of 52 cards and a gambler bets

that, it is a spade or an ace. What are the odds against his winning the
bet? [Ans.: 9:4]

10) The probability that a contractor will get a plumbing contract is 2/3
and the probability that he will not get an electrical contract is 5/9. If
the probability of getting at least 1 contract is 4/5, what is the
probability that he will get both the contracts? [Ans.: 14/45]

11) In a town of 6000 people 1200 are over 50 years old and 2000 are
female. It is known that 30% of the females are over 50 years. What is
the probability that a random chosen individual from the town is either
female or over 50 years? [Ans.: 13/30]

12) A Chartered Accountant applies for a job in two firms X and Y. He

estimates that the probability of his being selected in firm X is 0.7, and
being rejected at Y is 0.5 and the probability of at least one of his
application being rejected is 0.6. What is the probability be selected at
least one of the firms? [Ans.: 0.8]

13) If two dice are thrown, what is the probability that the sum of the
numbers on the dice is
(i) greater than 8, and (ii) neither 7 nor 11?
[Ans.: (i) 5/18, (ii) 7/9]

14) Suppose that 55% of all adults consume coffee, 45% regularly consume
carbonated soda, and 70% regularly consume at least one of these two
products.
21 | P a g e RAHUL STUDY GROUP

a) What is the probability that a randomly selected adult regularly

consume both coffee and soda? [Ans 0.3]
b) What is the probability that a randomly selected adult doesn’t
regularly consume at least one of them? [Ans 0.7]

15) The three most popular options on a certain type of new car are a
built – in GPS(A), a sunroof(B) and an automatic transmission(C). If
40% of all purchasers request A, 55% request B, 70% request C, 63%
request A or B, 77% request A or C, 80% request B or C, and 85%
request A or B or C, then determine the probability of the following
events:

a) The next purchaser will request at least one of the three options.
[Ans .85]
b) The next purchaser will select none of the three options. [Ans .15]
c) The next purchaser will request only an automatic transmission and
not either of the other two options. [Ans 0.22]
d) The next purchaser will select exactly one of these three options.
[Ans .38]

16) Consider the two events A and B with P(A)=0.4 and P(B)= 0.7.
Determine the maximum and minimum Possible Values of P(A∩ B) and
condition under which each of the value attained. [Ans 0.4, 0.1]

17) If n letters are placed at random in n envelops, what is the probability

that exactly n-1 letters will be placed in the correct envelops? [Ans 0]

18) Let A, B and C are the Arbitrary events. Show that the probability that
exactly one of these three events will be
P(A) + P(B) +P(C) - 2P(A∩B) – 2P(A∩C) - 2P(B∩C) +3 P(A∩ 𝐵 ∩ 𝐶)

19) An insurance company offers four different deductible levels – none,

low, medium and high – for its homeowner’s policyholders and three
different levels – low, medium and high – for its automobile
policyholders. The accompanying table gives proportions for the
various categories of policyholders who have both types of insurance.
22 | P a g e RAHUL STUDY GROUP

Homeowner’s
N L M H

L 0.04 0.06 0.05 0.03

M 0.07 0.10 0.20 0.10

H 0.02 0.03 0.15 0.15

Suppose an individual having both types of policies is randomly selected.

a) What is the probability that the individual has a medium auto

deductible and a high homeowner’s deductible? [Ans 0.10]
b) What is the probability that the individual has a low auto deductible?
A low homeowner’s deductible? [Ans 0.18, 0.19]
c) What is the probability that the individual is in the same category for
both auto and homeowner’s deductibles? [Ans 0.41]
d) Based on your answer in part (c), what is the probability that the two
categories are different? [Ans 0.59]
e) What is the probability that the individual has at least one low
deductible level? [Ans 0.31]
f) Using the answer in part (c), what is the probability that neither
deductible level is low? [Ans 0.69]

20) If A, B and C are mutually exclusive and exhaustive events, and

1 2
𝑃(𝐴) = 𝑃(𝐵) 𝑎𝑛𝑑 𝑃(𝐵) = 𝑃(𝐶), 𝑓𝑖𝑛𝑑 𝑃(𝐴), 𝑃(𝐵) 𝑎𝑛𝑑 𝑃(𝐶).
2 3
[Ans.: 1/6, 1/3, 1/2]

21) A box contains six 40 – W bulbs, five 60 – W bulbs, and four 75 – W

bulbs. If bulbs are selected one by one in random order, what is the
probability that at least two bulbs must be selected to obtain one that is
rated 75 – W? [Ans 0.6]

22) Suppose that a Balanced dice is rolled three times and let Xi denotes the
number that appear on the ith roll (i= 1 ,2 ,3 ). Evaluate P(X1>X2>X3)
[Ans 20/216]
23 | P a g e RAHUL STUDY GROUP

23) An Individual is presented with three different glasses of cola ,labeled C,

D and P. He is asked to taste all three and then list them in order of
preference. Suppose the same cola has actually been put into all three
glasses.
a. What are the simple events in the ranking experiment, and what
probability you assign to each one? [Ans 1/6]
b. What is the probability that C is ranked first? [Ans 0.333]
c. What is the probability that C is ranked first and D is ranked last?
[Ans 1/6]

24) Consider the type of clothes dryer (gas or electrical) purchased by each
of the five different customers at a certain store.

a. If the probability that at most one of these purchases an electrical

dryer is 0.428. What is the probability that at least two purchaser
an electrical dryer? [Ans .572]
b. If P (all five purchases gas)= 0.116 and P( all five purchases
electric)=0.005, What is the probability that at least one of each
type is purchased? [Ans 0.879]
COMBINATION AND PERMUTATION

25) A bag contains 3 white, 4 black & 5 red balls. Two balls are drawn one
by one. Find the probability that:
i) Both are red
ii) First is red & second is black,
iii) One is red & other is black
[Ans.: (i) 5/33, (ii) 5/33, (iii) 10/33]

26) A box of nine golf gloves contains two left - handed and seven rights -
handed gloves
(i) If two, gloves are randomly selected from the box without
replacement, what is the probability that (a) both gloves are right
handed and (b) one is left - handed and one right - handed glove?
(ii) If three gloves are selected without replacement, what is the
probability that all of them are left - handed?
(iii) If two gloves are randomly selected with replacement, what is
the probability that they-would both be right - handed?
[Ans.: (i) 7/12, 7/18; (ii) 0; (iii) 49/81]
24 | P a g e RAHUL STUDY GROUP

27) A box in a certain supply room contains four 40-w light-bulb, five 60-w
bulb and six 75-w bulbs. Suppose that three bulbs are randomly
selected.
a) What is the probability that exactly two of the selected bulb are rated
75-w?
b) What is the probability that one bulb of each type is selected?
c) What is the probability that all three of the selected bulbs have the
same rating?
d) Suppose now that bulb are to be selected one by one until a 75-w
bulb is found. What is probability that it is necessary to examine at
least six bulb. [Ans. .2967, .0747, .2637, .0421]

28) In a lot of 12 microwave ovens, there are 3 defective units. A person has
ordered 4 of these units and since each is identically packed, the
selection will be random. What is the probability that i) all units are
good?, ii) exactly 3 units are good?, iii) at least 2 units are good?:
[Ans.: (i) 14/55, (ii) 28/55, (iii) 54/55]

29) A bag contains 5 white and 8 red balls. two drawings of 3 balls each are
made such that:
a. The ball are replaced before the second trial,
b. The balls are not replaced before the second trial.

Find the probability that the first drawing gives 3 white and the second
draw gives 3 red-balls in each case. [Ans.: (a) 140/20449 (b) 7/429]

30) A committee of four has to be formed from among 3 economists, 4

engineers, 2 statisticians and 1 doctor.
(i) What is the probability that each of the four professions is
represented on the committee?
(ii) What is the probability that the committee consists of the doctor
and at least one economist? [Ans.: (i) 4/35; (ii) 32/105]

31) A word consists of 9 letters, 5 consonants and 4 vowels. Three letters

are chosen at random. What is the probability that more than one
vowel will be selected? [Ans.: 17/42]
25 | P a g e RAHUL STUDY GROUP

32) A production facility employs 20 workers on the day shift, 15 workers

on the swing shift, and 10 workers on the graveyard shift. A quality
control consultant is to select 6 of these workers for – in depth
interviews. Suppose the selection is made in such a way that any
particular group of 6 workers has the same chance of being selected as
does any other group.

A. How many selections results in all 6 workers coming from the day
shift? What is the probability that all 6 selected workers will be
from the day shift? [Ans 38760, 0.0048]
B. What is the probability that all 6 selected workers will be from the
same shift? [Ans 0.0054]
C. What is the probability that at least two different shifts will be
represented among the selected workers? [Ans .9946]
D. What is the probability that at least one of the shifts will be
unrepresented in the sample of workers? [Ans 0.2285]

33) The computer of six faculty members in a certain department are to be

replaced. Two of the faculty members have selected laptop machine
and other four have chosen desktop machines. Suppose that only two
computers to be set up are randomly selected from the six (implyfying
15 equally likely)

a) What is the probability that both selected setups are for laptop
computer?
b) What is the probability that both setup are desktop machines.
c) What is the probability that at least one selected setup is for a
desktop computers.
d) What is the probability that at least one computer of each type is
chosen. [Ans. .067, .40, .933, .533]

34) A box contains 30 red balls, 30 white Balls, and 30 blue balls. If 10 balls
are selected at random, without replacement what is the probability
that at least one color will be missing from the selection?
(𝟔𝟎) (𝟑𝟎)
𝟏𝟎
[ Ans 3 − 𝟑 𝟏𝟎 ]
(𝟗𝟎) (𝟗𝟎)
𝟏𝟎 𝟏𝟎
26 | P a g e RAHUL STUDY GROUP

35) Fifteen telephones have just have been received at an authorized

service center. Five of these telephone is cellular, five are cordless and
the other five are corded phones. Suppose that these component are
randomly allocated the number 1, 2, 3….15 to establish the order in
which they will be serviced.

a. What is the probability that all cordless phones are among the
first ten to be serviced? [Ans .0839]
b. What is the probability that after servicing ten of these phones,
phones of only two of the three types remain to be serviced?
[Ans .2498]
c. What is the probability that two phones of each type are among
the first six serviced? [Ans .1998]

36) A car with 6 sparkplugs is known to have 2 malfunctioning ones. If 2

plugs are pulled out at random, what is the probability of getting at
least one malfunctioning plug? [Ans 0.6]

37) A bag contains N balls of which a are red. Two balls are chosen
randomly from the bag without replacement. Let 𝑝1 denote the
probability that the first ball is red and 𝑝2 the probability that 2nd ball is
red. Show that 𝑝1 =𝑝2

38) Your teacher knows 6 jokes and in each class tells 2 jokes; each joke has
an equal chance of being selected. What is the probability that, in a
given lecture, at least 1 joke is told that was not told in the previous
class? [Ans 28/30]

39) If 12 Balls are thrown at random into 20 boxes , What is the probability
that no box will receive more than one ball? [Ans 20!/8!x𝟐𝟎𝟏𝟐 ]

40) An Elevator in a building starts with five passengers and Stop at

seventh floors. If every Passenger is equally likely to get off at each floor
and all the passengers leave independently of each other , what is the
probability that no two passenger will get off the same floor?
[Ans 360/2401]
27 | P a g e RAHUL STUDY GROUP

41) Suppose that three runners from team A and three runners from team
B participate in a race. If all six runners have equal ability and there are
no ties, what is the probability that three runners from team B will
finish fourth, fifth and sixth? [Ans 1/20]

42) There are total K peoples, Find the probability that (i) K person have a
different birthday (ii) At least two of the people will have the same
𝟑𝟔𝟓𝐱𝟑𝟔𝟒𝐱……𝐱(𝟑𝟔𝟓−𝐤+𝟏)
birthday? [Ans (i) ,(ii) 1- (i) ]
𝟑𝟔𝟓𝒌

43) If k people seated in a random manner in a row containing n seats

(n>k), What is the probability that people will occupy k adjacent seats
(𝐧−𝐤+𝟏)!𝐤!
in the row? [Ans ]
𝐧!

44) Suppose that 35 people are divided in a random manner into two teams
in such a way that one team contains 10 people and the other team
contains 25 people. what is the probability that two particular people A
and B will be on the same team? [Ans .5798]

45) Suppose that 100 mathematic Students are divided into five classes,
each containing 20 Students and that awards are given to 10 of these
students. If each student is equally likely to receive an award, What is
the probability that exactly two students in each class will receive
award. [Ans .0143]

46) If seven balanced dice are rolled , what is the probability that each of
𝟕!
the six different numbers will appear at least once ? [Ans ]
𝟐(𝟔)𝟔

47) Suppose that a deck of 52 cards contains 13 red cards, 13 yellow ,13
blue cards ,13 blue cards and 13 green cards. If the 52 cards are
distributed in a random manner among four players in such a way that
each player receives 13 cards, what is the probability that each player
𝟒! (𝟏𝟑!)𝟒
receive 13 cards of the same color? [Ans ]
𝟓𝟐!
28 | P a g e RAHUL STUDY GROUP

48) If three letters are placed at random in three envelops, what is the
probability that exactly one letter will be placed in the correct
envelops? [Ans 1/2]

49) Individual A has a circle of five close friends (B, C, D, E and F). A has
heard a certain rumor from outside the circle and has invited the five
friends to a party to circulate the rumor to begin, A Selects one of the
five at random and tells the rumor to the chosen Individual. That
individual then selects at random one of the four remaining individuals
and repeat the rumor. Continuing, a new Individual is selected from
those not already having heard the rumor by the individual who has
just heard it, until everyone has been told.

a. What is the probability that the rumor is repeated in the order

B, C, D, E and F. [Ans 1/120]
b. What is the probability that F is the third person at the party to be
told the rumor? [Ans 0.2]
c. What is the probability that F is the last person to hear the
rumor? [Ans 0.2]
d. If at each stage the person who currently has the rumor does not
know who has already heard it and selects the next recipient at
random from all five possible individuals, what is the probability
that F has still not heard the rumor after it has been told ten times
𝟒 𝟏𝟎
at that party? [Ans ( ) ]
𝟓

50) 25 books are placed at random on a shelf. The probability that a

particular pair of books shall be always together? [Ans 2/25]

51) Suppose that each person out of a group of 4 friends is randomly

assigned to one of the 6 classes. What is the probability that no class
has more than one person from this group? [Ans 5/18]

52) The nine digits 1, 2, … 9 are arranged in random order to form a nine
digit number, which uses each digit exactly once. Find the probability
that 1, 2 and 3 appear as neighbor in the increasing order? [Ans 1/72]
29 | P a g e RAHUL STUDY GROUP

53) Twelve balls are distributed at random among three boxes. what is
𝟏𝟐∁𝟑 𝑿 𝟐𝟗
probability that the first box will contain 3 balls? Ans ( )
𝟑𝟏𝟐

54) A stereo store is offering a special price on a complete set of

components (receiver, compact disc player, speakers, and turntable). A
purchaser is offered a choice of manufacturer for each component:

Receiver: Kenwood, Onkyo, pioneer, Sony, Sherwood

Compact disc player: Onkyo, pioneer, Sony, Technics
Speakers: Boston, Infinity, Polk
Turntable: Onkyo, Sony, Teac, Technics

A switchboard display in the store allows a customer to hook together any

selection of components (consisting of one of each type). Use the product
rules to answer the following questions:

a) In how many ways can one component of each type be selected?

[Ans 240]
b) In how many ways can components be selected if both the receiver
and the compact disc players are to be Sony? [Ans 12]
c) In how many ways can components be selected if none is to be Sony?
[Ans 108]
d) In how many ways can a selection be made if at least one Sony
component is to be included? [Ans 132]
e) If someone flips switches on the selection in a completely random
fashion, what is the probability that the system selected contains at
least one Sony component? Exactly one Sony component?
[Ans .55, 0.413]

INDEPENDENT EVENTS
1 7
55) Events A and B are such that 𝑃(𝐴) = , 𝑃(𝐵) = and
2 12
1
P(not A or Not B) = . State whether A & B are (i) mutually exclusive,
4
(ii) Independent [Ans. no, no]
30 | P a g e RAHUL STUDY GROUP

56) Let A and B be two independent events. The probability of their

simultaneous occurrence is 1/8 and this- probability that neither
occurs is 3/8. Find P(A) and P(B). [Ans.: 1/4, 1/2]

57) Anil and Rajesh appear in an interview for two vacancies. The
probability of Anil's selection is 1/3 and that of Rajesh's selection is
1/5. Find the probability that
(a) None will be selected;
(b) Only one of them will be selected;
(c) Both of them will be selected;
(d) At least one of them will be selected
[Ans.: (a) 8/15, (b) 2/5, (c) 1/15, (d) 7/15]

58) The odds against A solving a problem are 10 to 7 and the odds in favour
of B solving the problem are 15 to 12. What is the probability that, if
both of them try, the problem will be solved? [Ans.: 113/153]

59) A husband and a wife appear in an interview for two vacancies in the
same post. The probability of husband's selection is 1/7 and that of
wife's selection is 1/5. What is the probability that (a) both of them will
be selected; (b) only one of them will be selected, and (c) none of them
will be selected. [Ans.: (a) 0.029, (b) 0.286, (c) 0.686]

60) A certain team wins with probability 0.7, loses with probability 0.2 and
ties with probability 0.1. The team plays three games. Find the
probability that the team wins at least two of the games, but not lose.

[Ans.: 0.49]

61) Three critics review a book. Odds in favor of the book are 5:2, 4:3 and
3:4 respectively for the three critics. Find the probability that the
majority is in favor of the book. [Ans.: 209/343]

62) 70% of all vehicles examined at a certain emissions inspection station

pass the inspection. Assuming that successive vehicles pass or fail
independently of one another, calculate the following probabilities:

a) P (all of the next three vehicles inspected pass) [Ans 0.343]

31 | P a g e RAHUL STUDY GROUP

b) P (at least one of the next three inspected fails) [Ans 0.657]
c) P (exactly one of the next three inspected passes) [Ans 0.189]
d) P (at most one of the next three vehicles inspected passes)
[Ans 0.216]

63) A machine operates if all its three components function. The probability
that the first component fails during the year is 0.14, the second
component fails is 0.10 and the third component fails is 0.05. What is
the probability that the machine will fail during the year? [Ans.: 0.2647]

64) Suppose that 10000 tickets are sold in one lottery and 5000 tickets are
sold in another lottery. if a person owns 100 tickets in each lottery,
what is the probability that she will win at least one first prize.
[Ans .0298]

65) Consider an experiment in which a fair coin is tossed until a head is

obtained for the first time. if this experiment is performed three times.
what is the probability that exactly the same number of tosses will be
required for each of the three performances? [Ans 1/7]

66) Two boys A and B throw a ball at target. Suppose that A will hit the
Target on any throw is 1/3 and the probability that B will hit the target
on any throw is ¼. Suppose that A will throw the first and the two boys
take turn throwing. Determine the probability the target will be hit for
the first time on the third throw of boy A. [Ans 1/12]

67) Suppose that A, B and C are three events such that A and B are disjoint
events, A and C are independent, and B and C are independent. Suppose
also that 4P(A) =2P(B) =P(C)>0 and P(AUBUC) =5P(A). Determine
the Value of P(A) [Ans 1/6]

68) Suppose that three red balls and three white balls are thrown at
random into three boxes and that throws are independent. What is the
probability that each box contains one red and one white ball?
𝟐 𝟐
[Ans ( ) ]
𝟗
32 | P a g e RAHUL STUDY GROUP

69) A boiler has five identical relief valves. The probability that any
particular valves will open on demand is 0.95. Assuming independent
operation of the valves, Calculate P (at least one valves open) and
P (at least one valve fails to open)? [Ans .99999, .2262 ]

70) Two pumps connected in parallel fail independently of one another on

any given day. the probability that only the older pump will fail is 0.10.
and the probability that only the newer pump will fail on any given day
is 0.05. what is the probability that both the pumping system will fail on
any given day? [ Ans .0059]

71) One satellite is scheduled to be launched from India and another is

launching from Nepal. let A denotes the event that the Nepal launches
goes of the schedule and B be the events that India Launch goes off the
scheduled. If A and B are independent events with P(A) > P(B)
P(AUB)= 0.626, P(AՌB) =0.144. Determine the value of P(A) and P(B).
[Ans 0.45, 0.32]

72) Let X be the set of positive integers denoting the number of tries it
takes the Indian cricket team to win the world cup. The team has equal
odds for winning or losing any match. What is the probability that they
will win in odd number of matches? [Ans 2/3]

CONDITIONAL PROBABAILITY

73) For any two events A and B with P(B)>0,

Prove that P(A’/B) =1 – P(A/B)

74) For any three Events A, B and D Such that P(D)>0, Prove that
P(A𝑈𝐵|𝐷)= P(A|D) +P(B|D) -P(A∩ 𝐵|𝐷).

75) Suppose that A and B are the events such that P(A)=1/3, P(B)=1/5 ,
P(A/B)+P(B/A)=2/3..find P(A’UB’) [Ans 11/12]

76) Suppose A and B are Independent Events Such that P(A)=1/3 and
P(B)>0. What is the value of P(AUB’/B). [Ans P(A)]
33 | P a g e RAHUL STUDY GROUP

1 1
77) A and B are two events such that 𝑃(𝐴) = , 𝑃(𝐵) = , 𝑃(𝐴 ∪ 𝐵) =
4 3
1 𝐵 1
. 𝐻𝑒𝑎𝑛𝑐𝑒, 𝑃 ( ) = 𝐶𝑜𝑚𝑚𝑒𝑛𝑡. [Ans.: False]
2 𝐴 5.

78) Among the examinees in an examination 30%, 35% and 45% failed in
Statistics, in Mathematics and in at least one of the subjects
respectively. An examinee is selected at random. Find the probabilities
that
(i) He failed in Mathematics only;
(ii) He passed in Statistics If it is known that he failed in
Mathematics. [Ans.: (i) 0.15, (ii) 3/7]

79) The probability that a trainee will remain with a company is 0.8. The
probability that an employee earns more than Rs. 20,000 per year is
0.4. The probability that an employee who was a trainee remained with
the company or who earns more than Rs. 20,000 per year is 0.9. What is
the probability that an employee earns more than Rs. 20,000 per year
given that he is a trainee who stayed with the company? [Ans.: 3/8]

80) The probability that a person stopping at a petrol pump will get his
tyres checked is 0.12, the probability that the will get his oil checked is
0.29 and the probability that he will get both checked is 0.07.
(i) What is the probability that a person stopping at this pump will
have neither his tyres checked nor oil checked?
(ii) Find the probability that a person, who has oil checked, will also
have his tyres checked. [Ans.: (i) 0.66, (ii) 7/29]

81) A box of 100 gaskets contains 10 gaskets with type A defect, 5 gaskets
with type B defects and 2 gaskets with both types of defect. Find the
probabilities that:
(i) A gasket to be drawn has a type B defect under the condition that
it has a type A defect, and
(ii) A gasket to be drawn has no type of defect under the condition
that it has no type A defect. [Ans.: (i) 1/5, (ii) 87/90]

82) Two factories manufacture the same machine parts. Each part is
classified as having either 0, 1, 2 or 3 manufacturing defects. The joint
probability for this is given below:
34 | P a g e RAHUL STUDY GROUP

Manufacturer Number of Defects

0 1 2 3

X 0.1250 0.0625 0.1875 0.1250

Y 0.0625 0.0625 0.1250 0.2500

(i) A part is observed to have no defect. What is the probability that it was
produced by X manufacturer?

(ii) A part is known to have been produced by manufacturer X. What is the

probability that the part has no defects?

(iii) A part is known to have two or more defects. What is the probability
that it was manufactured by X? (iv) A part is known to have one or more
defects. What is the probability that it was manufactured by Y?

[Ans.:(i) 0.67, (ii) 0.25, (iii) 0.455, (iv) 0.5385]

83) Each Contestant on a quiz show is asked to specify one of the six
possible categories from which question will be asked? Suppose
P(contestant request quiz i)=1/6 and the successive contestant choose
their category independently of one another .if there are three
contestant on a particular show selects different categories , what is the
probability that exactly one has selected category 1 ? [Ans 0.5]

84) Suppose that we classify all households into one of two states rich and
poor. The probability of a particular generation being in either of these
states depends only on states in which their parents were. If a parent is
poors today, their child is likely to be poor with a probability 0.7. If a
parent is rich today, their child is likely to be poor with probability 0.6.
What is the probability that the great grandson of a poor man will be
poor? [Ans 0.67]

85) An oil exploration company currently has two projects, one in the asia
and other in Europe. Let A be the events that Asian Project is successful
and B be the event that European project is successful. Suppose A and B
are independent events with P(A)= 0.4, P(B)=0.7.
35 | P a g e RAHUL STUDY GROUP

a. If the asian project is not successful, what is the probability that

the European project is also not successful? [Ans 0.3]
b. What is the probability that at least one of the two projects will be
successful? [Ans 0.82]
c. Given that at least one of the two projects is successful, what is the
probability that only the Asian Project is successful. [Ans 0.146]

86) Suppose A and B Are Independent Events Such that P(A)=1/3 and
P(B)>0. What is the value of P(AUB’/B). [Ans P(A)]
BAYES THEOREM

87) In a bolt factory, machines A, B, C manufacture respectively 25%, 35%

and 40% of the total. Of their output 5, 4, 2 per cent are known to be
defective bolts. A bolt is drawn at random from the product and is
found to be defective. What are the probabilities that it was
manufactured by; (i) Machine A, (ii) Machine B or C.
[Ans.: (i) 0.36; (ii) 0.64]

88) If a machine is correctly set-up, it produces 90% acceptable items. If it

is incorrectly set up, it produces only 40% acceptable items. Experience
shows that 80% of the set ups are correctly done.
(i) If after a certain set up, out of the first two items produced, first is
found to be acceptable and second unacceptable, what is the
probability that the machine is correctly set-up?
(ii) If the machine produced first two items acceptable, what is the
probability that the machine is correctly set-up?
[Ans.: (i) 0.6, (ii) 0.953]

The results of an investigation by an expert on a fire accident are

summarized below:

(i) Probability (there could have been a short, circuit) = 0.8.

(ii) Prob. (LPG cylinder explosion) = 0.2.
(iii) Chance of fire accident is 30% given a short circuit and 95%
given an LPG explosion.
Based on these, what do you think is the most probable cause of fire?
[Ans.: P(E1/X) = 24/43, P(E2/X) = 19/43; so s/circuit]
36 | P a g e RAHUL STUDY GROUP

89) A man has five coins, one of which has two heads. He randomly takes
out a coin and tosses it three times:
i) What is the probability that it will fall head upward all the limes?
ii) If it always falls head upward, what is the probability that it is the
coin with two heads? [Ans.: (i) 0.3, (ii) 2/3]

90) A Box contain three coins with a head on each side, four coins with tail
on each side and two fair coins, if one of the nine coins selected at
random and tossed once, what is the probability that a head will be
obtained? [Ans 4/9]

91) A transmitter is sending a message by using a binary code, namely a

sequence of 0’s and 1’s. Each transmitted bit (0 or 1) must pass through
three relay to reach the receiver. At each relay, the probability is 0.20
that the bit sent will be different from the bit received (a reversal).
Assume that the relays operate independently of one another.

Transmitted → Relay 1 → Relay 2 → Relay 3 → Receiver

a. If a 1 is sent from the transmitter, what is the probability that a 1 is

sent by all three relays? [Ans 0.512]
b. If a 1 is sent from the transmitter, what is the probability that 1 is
received by the receiver? [Ans 0.608]
c. Suppose that 70 % of all bits sent from transmitter are 1’s. If a 1 is
received by the receiver, what is the probability that a 1 was sent?
[Ans 0.7835]

92) A chemical engineer is interested in determining whether a certain

trace impurity is present in a product. An experience has a probability
of 0.8 of detecting the impurity if it is present. The probability of not
detecting the impurity if it is absent is 0.9. The prior probability of the
impurity being present and being absent are 0.4 and 0.6, respectively.
Three Separate experiment result in only two detections. what is the
posterior probability that the impurity is present? [Ans 0.905]

93) Population of a city is 40% male and 60% female. Suppose also that
50% of males and 30% of females in the city smoke. The probability
that a smoker in the city is male? [Ans 10/19]
37 | P a g e RAHUL STUDY GROUP

94) Consider a railway signaling system: a signal is received by station A

from the traffic control office and then transmit it to station B. Suppose
that the origin (Traffic control office) signal can be yellow or red with
probability 4/5 and 1/5 respectively. The probability of each station
receiving the signal correctly from its predecessor is ¾. If the signal
received at station B is yellow, then the probability that the original
signal was yellow? [Ans 20/23]

95) A physical therapist at Enormous State university known that football

team will play 40% of its games an artificial turf for this season. He also
knows that football player’s chances of incurring a knee injury are 50%
higher if he is playing on artificial truf instead of grass. If player’s
probability of knee injury on artificial turf is 0.42. What is the prob.
a) A randomly selected football player incurs a knee injury.
b) A randomly selected football player with knee injury incurred the
injury playing on a grass? [Ans. .336, .5]

96) A Crime is committed by one of two suspects A and B. Initially there is

equal evidence against both of them. In further investigation at the
crime scene, it is found that the guilty party had a blood type found in
10% of the population. Suspect A does match this blood type, whereas
the blood type of type Suspect B is unknown, given this new
information, what is the probability that A party is guilty? [Ans 10/11]
Miscellaneous

1. Suppose that a balanced die is rolled repeatedly until the same number
appear on two successive rolls, and Let X denotes the number of that
are required. Determine the value of P(X=x) for x= 2 , 3 ,……
𝟓 𝟏
[Ans ( )𝒙−𝟐 ( ) ]
𝟔 𝟔

2. A particular airline has 10 A.M flights from Chicago to New York,

Atlanta and Los Angeles. Let A denotes the event that the New York
flights is full and define Event B and C analogously for the other two
flights. Suppose P(A)=0.6, P(B) = 0.5, P(C)=0.4 and three events are
independent. What is the probability that:
a. All three flights are full? That at least on flight is not full?
[Ans 0.12, 0.88]
38 | P a g e RAHUL STUDY GROUP

b. Only the New York flight is full? that exactly one of the three
flights is full? [Ans 0.18, 0.38]

3. Three groups of workers contain 3 men and 1 woman, 2 men & 2

women & 1 man and 1 woman respectively. One worker is selected at
random from each group, what is the probability that the group
selected consists of 1 man and 2 women. [Ans.: 5/16]

4. A box has 10 red balls and 5 black balls. A ball is selected from the box.
If the ball is red, it is returned to the box. If the ball is black, it and 2
additional black balls are added to the box. The probability that a
second ball selected from the box will be red? [Ans 98/153]

5. A particular men’s competition has unlimited number of rounds. In

each round, every participant has to complete a task. The probability of
a participating completing the task in a round is p. If a participant fail to
complete a task in a round, he is eliminated from the competition. He
participates in every round before being eliminated. The competition
begins with 3 participants. What is the probability that all 3
(𝟏−𝒑)𝟑
participants are eliminated in the same round? [Ans ]
𝟏−𝒑𝟑

6. Three married couples sit down at a round table at which there are 6
chairs. All if thee possible seating arrangements of the 6 people are
equally likely. The probability that each husband sits next to his wife?
[Ans 2/15]

7. Each of 4 economists is asked to prescribe one of the 4 economic

policies. Each economic is equally likely to prescribe any of the 4
different policies. What is the probability that each of the economists
prescribes the different policy? [Ans 3/32]

8. There are 5 women on the platform of a train station. The train that
they are waiting for has 3 coaches and each of them is equally likely to
enter any coach. What is the probability that they will enter the same
coach? [Ans 1/25]
39 | P a g e RAHUL STUDY GROUP

9. ICICI Bank collects data on 10000 respondents. Out of the 6800 men,
4200 have credit cards and out of the 3200 women, 2500 have credit
cards. Out of the men with credit cards, 1200 have unpaid balance,
whereas out of the women with credit cards, 1400 have unpaid balance.
What is the probability that an individual selected at random is a man
with no unpaid balance? [Ans 0.56]

10. Amit has a box containing 6 red balls and 3 green balls. Anita has a box
containing 4 red balls and 5 green balls. Amit randomly draws one ball
from the box and put it into Anita’s box. Now Anita randomly draws one
ball from her box. What is the probability that the balls drawn by Amit
& Anita were of different colors? [Ans 7/15]

11. Two patients share a hospital room for 2 days. Suppose that on any
given day, a person independently picks up a airborne infection with a
1
probability of . An individual who is infected on the first day will
4
certainly pass it to the other patient on the second day. Once
contracted, the infection stays for at least 2 days.

a. What is the probability that both patients have contracted the

infection by the end of the second day? [Ans 121/256]
b. What is the probability that fewer than two patients have the
infection by the end of the second day? [Ans 135/256]

12. A survey of asset ownership in poor household in rural UP and Bihar

found that 40% of the household own a radio, 15% own a television &
60% owns a cycle. It also found that 5% of the households own both
radio and a television, 26% own both a radio and a bicycle, 5% own
both a television and a bicycle and 1% all the three. If a randomly
selected poor household in these areas found to own exactly one of
these 3 assets, what is the probability that it is a bicycle? [Ans 15/23]

13. Jai and Vijay are taking statistics. The exam has only 3 grades A, B and
C. The probability that Jai gets a B is 0.3, the probability that Vijay gets a
B is 0.4, the probability that neither gets an A, but at least one gets B is
40 | P a g e RAHUL STUDY GROUP

0.1. What is the probability that neither gets a C but at least one gets a
B? [Ans 0.6]

14. A number𝑋1 , is chosen at random from the set (1, 2). Then a number𝑋2
is chosen at random from the set (1, 𝑋1 ). The probability that 𝑋1 = 2,
given that 𝑋2 =1? [Ans 1/3]
15. Four taste testers are asked to independently rank three different
brands of chocolate (A, B, C). The chocolate each tester likes best is
given the rank 1, then 2 and then 3. After this, the assigned ranks for
each of the chocolates are summed across the testers. Assuming the
testers cannot really discriminate between the chocolates, so that each
is assigning her rank at random. The probability that chocolate A
receives a total sum of 4? [Ans 1/81]

16. An insurance policy –holder can submit upto 5 claims. The probability
that the policyholder submits exactly n claims is 𝑝𝑛 , for n = 0, 1, 2, 3, 4,
5. If is known

a) The difference b/w 𝑝𝑛 𝑎𝑛𝑑 𝑝𝑛+1 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

b) 40% of the policyholders submit 0 or 1 claim.

What is the probability that a policyholder submits 4 or 5 claims?

[Ans 0.26]

17. A hospital determines that N, the number of patients in a week, is a

random variable with P[N=n]=2−𝑛−1 , 𝑛 ≥ 0. The hospital also
determines that the number of patients in a given week is independent
of the number of patients in any other week. The probability that there
are exactly 7 patients during a two-week period? [Ans 1/64]

18. A student is answering a multiple-choice examination. Suppose a

question has m possible answers. The student knows the correct
answer with probability p. if the student knows the correct answer, she
picks it with probability 1: otherwise, she picks randomly from the
choices with probability 1/m each. Give that the student picked the
correct answer, the probability that she knew the correct answer is
[Ans mp/[1 + (m – 1) p]
41 | P a g e RAHUL STUDY GROUP

19. There are 4 married couples in a club. A 3-member committee must be

formed from among them, such that no married couple is part of the
committee. Find the probability in which this committee can be formed
? [Ans 4/7]

20. Suppose that 80% of all statisticians are shy, whereas only 15% of all
economists are shy. Suppose also that 90% of the people at a large
gathering are economists and the other 10% are statisticians. If you
meet a shy person at random at the gathering, what is the probability
that the person is a statistician? [Ans 80/215]

21. Three Players A, B and C take turns Playing a game as follows. A and B
play in the first round. The winner plays C in the second round, while
the looser sits out. The game continuing in the fashion, with the winner
of the current round playing the next round with the person who sits
out the current round. The Game end ends if the players wins twice in
succession; This player declare the winner of the contest. For any of the
rounds assume that two players playing the round each have a
probability ½ of winning the rounds, regardless how the past rounds
were won or lost

a. The probability that A becomes the winner of the contest?

[Ans 5/14]
b. The probability that C becomes the winner of the contest?
[Ans 2/7]
c. The probability that the game continues indefinitely, with no one
winning twice in succession [Ans 0]

22. Mr. A and B are independently tossing a coin. Their coins have a
probability 0.25 coming head. After each of them tossing the coin twice,
we see a total of two heads. What is the probability that Mr. A had
exactly one head? [2/3]

23. Consider the square with vertices A, B, C and D. Call a pair of vertices in
the square adjacent if they are connected by an edge in the figure. You
may have four color RED, BLUE, GREEN, YELLOW. Find the probability
of coloring the vertices A,B, C D such that no adjacent vertices share the
same color? [0.328125]
42 | P a g e RAHUL STUDY GROUP

24. A slip of paper is given to person A, who marks it with either (+) or (-).
The probability of her writing (+) is 1/3. Then, the slip is passed
sequentially to B, C and D. Each of them either changes the sign on the
slip with probability 2/3 or leaves it as it is with probability 1/3.

a. Compute the probability that the final sign is (+) if A wrote (+).
[Ans 13/27]
b. Compute the probability that the final sign is (+) if A wrote (-).
[Ans 14/27]
c. Compute the probability that A wrote (+) if the final sign is (+).
[Ans 13/41]

25. A box contains four red balls, five white balls and six blue balls. Suppose
that three balls are drawn randomly

a. What is the probability three of the selected balls are of the same
color? [Ans 34/455]

b. If drawing a blue, what is the probability that at least 6 balls will

be drawn to obtaining a success? [Ans 0.0421]

26. Suppose a box contain five biased coins with probability of head as 0,
1/4, 1/2, 3/4, 1 respectively. One coin is selected at random and tossed
twice.
a. What is the probability of obtaining tail on the first toss? [Ans 0.5]
b. If a tail is obtained on first toss, what is the probability that
another tail will be obtained on the second toss? [Ans 0.75]

27. The letters of the word COCHIN are permuted and all the permutations
are arranged lexicographically (i.e., in alphabetical order as in an
English dictionary). Find the probability of words that appear before
the word COCHIN is [Ans 96/360]

28. For any positive integers k, ℓ with k ≥ ℓ, let C(k, ℓ) denote the number
of ways in which ℓ distinct objects can be chosen from k objects.
Consider n ≥ 3 distinct points on a circle and join every pair of points
43 | P a g e RAHUL STUDY GROUP

by a line segment. If we pick three of these line segments uniformly at

random, what is the probability that we choose a triangle?
[Ans C(n,3)/ C(C(n,2),3)]

CHAPTER -4
RANDOM VARIABLE
(PDF ,CDF ,ExPECTATION AND VARIANCE )
DISCRETE
1) Determine whether the given values can serve as the values of
probability distribution of the random variable with the range x=1,2, 3
and 4.
a) 𝑥 1 2 3 4
p(𝑥) 0.25 0.75 0.25 0.25 (Ans; No)
b) 𝑥 1 2 3 4
p(𝑥) 0.15 0.27 0.29 0.29(Ans; yes)

2) Determine whether the given function can serve as the probability

distribution of a random variable with the given range.
𝑥−2
a) 𝑓(𝑥) = 𝑥 = 1, 2, 3, 4, 5 (Ans; no)
5
𝑥2
b) 𝑓(𝑥) = 𝑥 = 0, 1, 2, 3, 4 (Ans; yes)
30
1
c) 𝑓(𝑥) = 𝑥 = 1, 2, 3, 4, 5(Ans; yes)
5

3) Determine the value of C so that function serve as prob. distribution

function.
a) 𝑓(𝑥) = 𝑐𝑥 𝑥 = 1, 2, 3, 4, 5 (Ans: 1/15)
2
b) 𝑓(𝑥) = 𝑐𝑥 𝑥 = 1, 2, 3,.....6 (Ans: 1/91)
44 | P a g e RAHUL STUDY GROUP

1 𝑥
c) 𝑓(𝑥) = 𝑐 ( ) 𝑥 = 1, 2, 3,..... (Ans: 3)
4

4) Find the CDF of the below PDF.

a) 𝑥 1 2 3 4
1 2 6 10
p(𝑥)
19 19 19 19

b) 𝑥 0 1 2
p(𝑥) 0.500 0.167 0.333

c) 𝑥 1 2 3 4
p(𝑥) 0.4 0.3 0.2 0.1

5) Find the CDF of the below prob. distribution function

1 𝑥
a) P(x= 𝑥) = {3 (4) 𝑥 = 1,2,3 …}
0 otherwise

𝟏 [𝒙]
[Ans F( 𝒙) = {𝟏 − (𝟒) 𝒙 = 𝟏, 𝟐, 𝟑 …}
𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞

(1 𝑥−1
b) P(x= 𝑥) = { − 𝑝) 𝑝 𝑥 = 1,2,3 … .}
0 otherwise

[𝒙]
Ans F(𝒙) ={(𝟏 − (𝟏 − 𝒑) ) 𝒙 = 𝟏, 𝟐, 𝟑 … .}
𝟎 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞

6) If X has the distribution function

0 𝑥<1
1⁄ 1≤𝑥≤4
3
F(𝑥) = 1⁄2 4≤𝑥<6
5⁄ 6 ≤ 𝑥 < 10
6
{ 1 𝑥 ≥ 10 }
45 | P a g e RAHUL STUDY GROUP

Find:

a) P(2 < 𝑋 ≤ 6) [Ans ½]

b) P(𝑋 = 4) [Ans 1/6]
c) PDF of x

7) If x has the distribution function

0 𝑥 < −1
1⁄ − 1 ≤ 𝑥 < 1
4
1
F(𝑥)= ⁄2 1 ≤ 𝑥 < 3
3⁄
4 3≤𝑥<5
{ 1 𝑥≥5 }
a) P(X≤ 3) [Ans ¾]
b) P(X = 3)[Ans ¼]
c) P(X < 3) [Ans ½]
d) P(𝑋 ≥ 1) [Ans ¾]
e) P(−0.4 < 𝑋 < 4) [Ans ¾]
f) P(X = 5)[Ans ¼]

8) A consumer organization that evaluates new automobiles customarily

reports the number of major defects in each car examined. Let X denote
the number of major defects in a randomly selected car of a certain
type. The cdf of X is as follows:
0 𝑥<0
. 06 0 ≤ 𝑥 < 1
. 19 1 ≤ 𝑥 < 2
. 39 2 ≤ 𝑥 < 3
F(x) =
. 67 3 ≤ 𝑥 < 4
. 92 4 ≤ 𝑥 < 5
. 97 5 ≤ 𝑥 < 6
{ 1 6 ≤𝑥
Calculate the following probability directly from the cdf:
a. P(2), that is P(X = 2) [Ans .20]
b. P(X > 3) [Ans .33]
c. P(2 ≤ 𝑋 ≤ 5) [Ans 0.78]
d. P(2 < X < 5) [Ans .53]
46 | P a g e RAHUL STUDY GROUP

9) Suppose that a random variable X has a discrete distribution with the

following p.f.
𝑐
, 𝑓𝑜𝑟 𝑥 = 1,2, … , ∞
𝑓(𝑥) = {2𝑥
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Find the value of the constant c. (Ans c=1/2)

10) Suppose that a random variable X has a discrete distribution with the
following p.f.
𝑐𝛳 𝑥
𝑓(𝑥) = { 𝑥 , 𝑓𝑜𝑟 𝑥 = 1,2, … , ∞
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
For What value of 𝛳 𝑎nd c is the above function is PMF?
𝟏
[Ans − , 0< 𝜭 <1 ]
𝐥𝐨𝐠 (𝟏−𝜭)

11) Suppose that a random variable X has a discrete distribution with the
following p.f.
2 𝑥 , 𝑓𝑜𝑟 𝑥 = 1,2, … , ∞
𝑓(𝑥) = {𝑐 ( )
3 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
For What value of - c is the above function is PMF? . (Ans c=1/2)

12) A contractor is required by a country planning dept. to submit one, two,

three, four, or five forms (depending on the nature of the project) in
applying for a building permit. Let Y = the number of forms required of
the next applicant. The probability that y forms are required is known
to be proportional to y – that is, p(y) = ky for y = 1, …, 5.

a) What is the value of k? (Ans k=1/15)

b) What is the probability that at most three forms are required? (Ans
0.4)
c) What is the probability that b/w two and four forms are required?
(Ans 0.2)
d) Could p(y) = 𝑦 2 /50 for y = 1, …, 5 be the pmf of Y? (Ans No)
47 | P a g e RAHUL STUDY GROUP

CONSTRUCTION OF PROBABILITY DISTRIBUTION

13) Find the Probability distribution of number of heads in four tosses of

a coin .

Ans 𝒙 0 1 2 3 4

p(𝒙) 1/16 4/16 6/16 4/16 1/16

14) From a lot of 30 bulbs which include 6 defectives , a sample of 4 bulb

is drawn at random with replacement . find the probability distribution
of the number of defective bulbs

Ans 𝒙 0 1 2 3 4

p(𝒙) 256/625 256/625 96/625 16/625 1/625

15) Many manufacturers have quality control programs that include

inspection of incoming material for defects. Suppose a computer
manufacturer receive computer board in lots of five. Two boards are
selected from each lot for inspection. Suppose two are defective boards.
a) Find the prob. dist. of defective boards.
b) Find CDF of x.

𝐀𝐧𝐬 𝒙 0 1 2

p(𝒙) 6/20 12/20 1/20

16) Two fair six – sided dice are tossed independently. Let M = the
maximum of the tosses.
a) What is the pmf of M?

m P(M=m)
48 | P a g e RAHUL STUDY GROUP

1 1/36

2 3/36

3 5/36

4 7/36

5 9/36

6 11/36

𝟎, 𝒎 < 1
𝟏
, 𝟏≤𝒎<2
𝟑𝟔
𝟒
, 𝟐≤𝒎<3
𝟑𝟔
𝟗
b) Determine the cdf of M and graph it. 𝑭(𝒎) = , 𝟑≤𝒎<4
𝟑𝟔
𝟏𝟔
,𝟒 ≤ 𝒎 < 5
𝟑𝟔
𝟐𝟓
,𝟓 ≤ 𝒎 < 6
𝟑𝟔
{ 1, 𝒎 ≥ 𝟔

17) After all students have left the classroom, a statistics professor notices
that four copies of the text were left under desks. At the beginning of
the next lecture the professor distributes the four books in a completely
random fashion to each of the four students (1, 2, 3 and 4) who claim to
have left book.

a) List all possible outcome.

b) Find pmf of x if x is the number of student who receives their own
book.

18) Some parts of California are particularly earthquake – prone. Suppose

that in one metropolitan area, 30% of all home –owners are insured
against earthquake damage. Four home – owners are to be selected at
random; let X denote the number among the four who have earth quake
insurance.

a) Find the probability distribution of X.

49 | P a g e RAHUL STUDY GROUP

𝑥 P(x=𝑥)

0 0.2401

1 0.4116

2 0.2646

3 0.0756

4 0.0081

b) Draw the corresponding probability histogram.

c) What is the most likely value for X? (Ans ;for x = 1)
d) What is the probability that at least two of the four selected have
earthquake insurance? (Ans 0.3483)

19) An automobile service facility specializing in engine tune–ups knows

that 45% of all tune-ups are done on four cylinder automobiles, 40% on
six-cylinder automobiles, and 15% on eight-cylinder automobiles.
Let X = the number of cylinder on the next car to be tuned.

a) What is the pmf of X?

b) Draw both a line graph and a probability histogram for the pmf of
part (a).
c) What is the probability that the next car turned has at least six
cylinders? More than six cylinders? [Ans (c).55 , .15]

20) Three couples and two single individuals have been invited to an
investment seminar and have agreed to attend. Suppose the probability
that any particular couple or individual arrives late is .4 (a couple will
travel together in the same vehicle, so either both people will be on
time or else both will arrive late). Assume that different couples and
individuals are on time or late independently of one another.
Let X=number of people who arrive late for the seminar.
50 | P a g e RAHUL STUDY GROUP

a) Determine the probability mass function of X. [Hint: label the three

couples #1, #2, and #3 and the two individuals #4 and #5.]
b) Obtain the cumulative distribution function of X, and use it to
calculate P(2≤ 𝑋 ≤ 6).

21) Airlines sometimes overbook flights. Suppose that for a plane with 50
seats, 55 passengers have tickets. Define the random variable Y as the
number of ticketed passengers who actually show up for the flight. The
probability mass function of Y appears in the accompanying table.

y 45 46 47 48 49 50 51 52 53 54
55

p(y) .05 .10 .12 .14 .25 .17 .06 .05 .03 .02
.01
51 | P a g e RAHUL STUDY GROUP

a) What is the probability that the flight will accommodate all ticketed
passengers who show up? [Ans .83]
b) What is the probability that not all ticketed passengers who show up
can be accommodated? [Ans .17]
c) If you are first person on the standby list (which means you will be
the first one to get on the plane if there are any seats available after
all ticketed passengers have been accommodated), what is the
probability that you will be able to take the flight? What is this
probability if you are the third person on the standby list? [Ans .27]

Random Variable, Expectation & Variance

22) A random variable X has the following probability distribution:

𝑥 1 2 4 8 16

P(X =𝑥): 0.05 .1 0.35 .4 0.10

Find E(X) and V(X) [Ans 6.45 ,15.6475]

23) An individual who has automobile insurance from a certain company is

randomly selected. Let Y be the number of moving violations for which
the individual was cited during the last 3 years. The pmf of Y is

Y 0 1 2 3

P(Y=y) 0.60 0.25 0.10 0.05

a) Compute E(Y). [Ans 0.60 ]

b) Suppose an individual with Y violations incurs a surcharge of $100𝑌 2 .
Calculate the expected amount of the surcharge. [Ans 110]

24) A random variable X has the following probability distribution:

𝑥 -2 -1 0 1 2 3

P(X =𝑥) 0.1 K 0.2 2K 0.3 K

i) Find the: value of K,

52 | P a g e RAHUL STUDY GROUP

ii) Find the expected value and .variance of X.

[Ans.: (i) 0.1, (ii) 0.8 and 2.16]

25) A random variable has the following probability distribution:

𝑥: 4 5 6 8

P(X=𝑥): 0.1 0.3 0.4 0.2

Find E[X - E(X)]2 [Ans.: 1.49]

26) A box contains 6 tickets. Two of the tickets carry a prize of Rs. 5 each,
the other four prizes are of Re. 1 each. If one ticket is drawn, what is the
expected value of the price? [Ans.: Rs. 2.33]

27) A company estimates the net profit on a new product, it, is launching, to
be Rs. 30 lakhs during the first year if it is successful, Rs 10 lakhs if it is
moderately successful, and a loss of Rs. 10 lakhs if it is unsuccessful.
The firm assigns the following probabilities to first year for the product
successful 0.15- moderately successful - 0.25. What are the expected
value and standard deviation of first year net profit for this product?

[Ans.: 1.00, 14.8]

28) A maker of soft drinks is considering the introduction of a new brand.

He expects to sell 50,000 to 1,00,000 bottles of its soft drink in given
period according to the following probability distribution.

No. of Bottles Sold (in Probability

‘000)

50 0.13

60 0.20

70 0.35

80 0.22

90 0.08

100 0.02
53 | P a g e RAHUL STUDY GROUP

If the product is launched, he will have to incur a fixed cost of Rs. 48,000.
However, each bottle sold would give him a profit of Rs. 1.25. Should he
introduce the new brand? [Ans.: Net Profit =Rs. 39,250, hence to be
introduced]

29) A small market orders copies of Magzine each week. Let X= demand
for magazine with pmf.

𝑥 1 2 3 4 5 6

P(X=𝑥) 1⁄ 2⁄ 3⁄ 4 3 2
15 15 15 ⁄15 ⁄15 ⁄15
Suppose that the store owner actually pays $2.00 for each copy if the
magazine and the price to customer is $4.00. if the magazines left at the end
of the week have no salvage values, is it better to order three or four copies
of the magazine. [Ans 4 Magzine]

30) Let X be the damage incurred in a certain type of accident during a

given year. Possible X values are 0, 1000, 5000 and 10000 with prob.
0.8, 0.1, 0.08 and 0.02. a particular company offers a $500 deducted
policy. If the company wishes its expected profit to be $100. What
premium amount should it charge. [Ans 700]

31) The probability distribution of a random variable X is given as:

𝑥: -2 3 1

P(X=𝑥): 1/3 ½ 1/6

Find E(2X + 5) and E(X2). [Ans.: 7, 6]

32) A random variable X has the following probability distribution:

𝑥; 0 1 2 3

P(X = 𝑥): 1/3 ½ 0 1/6

Find

i) Var(X),
54 | P a g e RAHUL STUDY GROUP

ii) Var(Y) where Y = 2X – 1 [Ans.: Var(X) = 1 and Var(Y) = 4]

33) An appliance dealer sells three different models of upright freezers

having 13.5, 15.9 and 19.1 cubic feet of storage space, respectively. Let
X = the amount of storage space purchased by the next customer to buy
a freezer. Suppose that X has pmf

𝑥 13.5 15.9 19.1

p(X=𝑥) 0.2 0.5 0.3

a) Compute E(X), E(𝑋 2 ) and V(X). [Ans 16.38 , 272.298 , 3.9936 ]

b) If the price of a freezer having capacity X cubic feet is 25X – 8.5, what
is the expected price paid by the next customer to buy a freezer?[Ans
401]
c) What is the variance of the price 25X – 8.5 paid by the next
customer? [Ans 2496 ]
d) Suppose that although the rated capacity of a freezer is X, the actual
capacity is h(X) = X – 0.01𝑋 2 . What is the expected actual capacity of
the freezer purchased by the next customer? [Ans 13.66 ]

34) The n candidate for a job have been ranked 1, 2, 3,…, n. Let X = the rank
of a randomly selected candidate, so that X has pmf

1
𝑥 = 1,2,3, . . , 𝑛
p(𝑥) ={𝑛
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Compute E(X) and V(X) using the shortcut formula.
𝒏+𝟏 𝒏𝟐 −𝟏
[Ans , ]
𝟐 𝟏𝟐

35) Suppose that the number of plants of a particular type found in a

rectangular sampling region in a certain geographic area is an rv X with
pmf

𝑐
𝑥 = 1, 2, 3, …
P(X=𝑥) = {𝑥 3
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
55 | P a g e RAHUL STUDY GROUP

Is E(X) finite? Justify your answer. [Ans Yes, it is finite)

36) Let X = the outcome when a fair die is rolled once. If before the die is
rolled you are offered either (1/3.5) dollars or h(X) = 1/X dollars,
would you accept the guaranteed amount or would you gamble?
[Ans. Accept the gamble]

37) A chemical supply company currently has in stock 100 lb of a certain

chemical, which it sells to customers in 5 – lb batches. Let X = the
number of batches ordered by a randomly chosen customer, and
suppose that X has pmf

𝑥 1 2 3 4

P(X=𝑥) 0.2 0.4 0.3 0.1

a) Compute E(X) and V(X). [Ans 2.3 , 0.81 ]

b) Then compute the expected number of pounds left after the next
customer’s order is shipped and the variance of the number of
pounds left? [Ans 88.5 ,20.25 ]

38) Suppose that one word is to be selected at random from the sentence
“THE GIRL PUT ON HER BEAUTIFUL RED HAT”. If x denotes the
number of letters in the word that is selected, what is the value of E(x)
and V(X)? [Ans 3.75 ,67/16]

39) Suppose that X is a random variable for which E(X)= 𝜇 and var(X)= 𝜎 2 .
Show that
E[X(X-1)]=𝜇 (𝜇 − 1) + 𝜎 2 ?

40) Suppose E(X) = 5 and E[X(X – 1)] = 27.5, then compute the following:

a) E(𝑋 2 ) [Ans 32.5 ]

b) V(X) [ Ans 7.5 ]
56 | P a g e RAHUL STUDY GROUP

Continuous probability Distribution

41) Continuous Random Variable X with PDF.

2)
𝑓(𝑥) = {0.09375(4 − 𝑥 − 2 ≤ 𝑥 ≤ 2}
0 otherwise
a) Verify the PDF function.
b) Compute P(X> 0) [Ans 0.5]
c) Compute P(-1< X < 1) [Ans 0.6875]
d) Compute P(X > 0.5) [Ans 0.6328]

42) The actual tracking weight of a stereo cartridge can be regarded as a

continuous rv X with pdf and the prescribe weight is 3.

(𝑥 − 3)2 ] 2 ≤ 𝑥 ≤ 4
ƒ(𝑥) = {𝑘 [1 –
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝟑
a) Find the value of k. [Ans ]
𝟒
b) Sketch the graph of ƒ(x).
c) What is the probability that the actual tracking weight is greater
than the prescribed weight? [Ans 0.5]
d) What is the probability that the actual weight is within .25 g of the
prescribed weight? [Ans 0.367]
e) What is the probability that the actual weight differs from the
prescribed weight by more than .5? [Ans 0.313]

43) Suppose that the pdf of the random variable x is as follows:

1
(9 − 𝑥 2 ) , −3 ≤ 𝑥 ≤ 3
f(𝑥)={36
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
i) P(X< 0) [Ans ½]
ii) P(−1 ≤ 𝑋 ≤ 1) [Ans 13/27]
iii) P(X > 2) [Ans 2/27 ]

44) An ice cream seller takes 20 gallons of ice cream in her truck each day.
Let x stand for the number of gallons that she sells. The probability is 0.1
that x=20. If she doesn’t sell all 20 gallons, the Distribution of x follows a
57 | P a g e RAHUL STUDY GROUP

continuous distribution with a pdf of the form

𝑐𝑥 , 0 < 𝑥 < 20
f(𝑥)-={
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

find the value of c such that P(X< 20) = 0. 𝟗 [Ans c=9/2000

]

45) Suppose that the pdf of the of the random variable x is as follows:
𝑐𝑥 2 , 1 ≤ 𝑥 ≤ 2
f(x)= {
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
i) Find the value of c such that this pdf is defined? [Ans c=3/7 ]
3
ii) P(x > ) [Ans 37/56 ]
2

46) Suppose that the pdf of the random variable x is as follows:

1
𝑥 ,0 ≤ 𝑥 ≤ 4
f(x)={8
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1
i) find the value of t such that p(x ≤ 𝑡) = [Ans t=2]
4
1
ii) find the value of t such that p(x ≥ 𝑡) = [Ans t=2√𝟐 ]
2

47) Suppose that the pdf of a random variable x is as follows:

𝑐𝑒 −2𝑥 , 𝑥 > 0
f(x)={
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

i) find the value of the constant c [Ans c = -1 ]

ii) p(1< 𝑥 < 2) [Ans 𝒆 − 𝒆−𝟒 ]
−𝟐

48) The p.d.f. of the random variable X is given by

𝑐
𝑓𝑜𝑟 0 < 𝑥 < 4
𝑓(𝑥) = {√𝑥 }
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Find
58 | P a g e RAHUL STUDY GROUP

a) The value of c; [Ans 1/4 ]

1
b) P (𝑋 < ) [Ans ¼ ]
4
c) P(X > 1) [Ans ½]

49) Find the distribution function of the random variable X whose

probability density is given by.
𝑥 𝑓𝑜𝑟 0 < 𝑥 < 1
𝑓(𝑥) = {2 − 𝑥 𝑓𝑜𝑟 1 ≤ 𝑥 < 2}
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

𝟎, 𝒙<𝟎
𝒙𝟐
, 𝟎≤𝒙<𝟏
𝟐
[Ans F(𝒙)= ]
𝒙𝟐
𝟐𝒙 − − 𝟏, 𝟏 ≤ 𝒙 < 𝟐
𝟐
{ 𝟏, 𝒙≥𝟐

50) Find the distribution function of the random variable X whose

probability density is given by.
𝑥
𝑓𝑜𝑟 0 < 𝑥 ≤ 1
2
1
𝑓𝑜𝑟 1 ≤ 𝑥 < 2
𝑓(𝑥) = 2
3−𝑥
𝑓𝑜𝑟 2 < 𝑥 < 3
2
{ 0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 }
𝟎, 𝒙<𝟎
𝒙𝟐
, 𝟎≤𝑥<1
𝟒
𝟏
F(𝑥)= (𝟐𝒙 − 𝟏), 𝟏≤𝑥<2
𝟒
𝟏
(𝟔𝒙 − 𝒙𝟐 − 𝟓), 𝟐 ≤ 𝑥 < 3
𝟒
{ 𝟏, 𝒙≥𝟑

51) a. Sketch the graph of the function

1 1
f(𝑥)= { − ǀ𝑥 − 3ǀ ,if 1≤ 𝑥 ≤ 5
2 4
59 | P a g e RAHUL STUDY GROUP

0 , otherwise
b. Find F , the distribution of X and show that it is continuous

52) In commuting t work, a professor must first get on a bus near her house
and then transfer to a second bus. If the waiting time (in minutes) a at
each stop has a uniform distribution with A = 0 and B = 5, then it can
be shown that the total waiting time Y has the pdf
1
𝑦 0≤𝑦 <5
25
ƒ(𝑦) ={ 2 1
− 𝑦 5 ≤ 𝑦 ≤ 10
5 25
0 𝑦 < 0 𝑜𝑟 𝑦 > 10
a) Sketch a graph of the pdf of Y.
∞
b) Verify that ∫−∞ ƒ(𝑦)𝑑𝑦 = 1.
c) What is the probability that total waiting time is at most 3 min?
[Ans 0.18]
d) What is the probability that total waiting time is at most 8 min?
[Ans 0.92]
e) What is the probability that total waiting time is between 3 and 8 min
[Ans 0.74]
f) What is the probability that total waiting time is either less than 2
min or more than 6 min? [Ans 0.4]

53) Suppose that the pdf of the random variable x is as follows:

𝑐
𝑓𝑜𝑟 0 < 𝑥 < 1
f(x)={(1−𝑥)1/2
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
i) find the value of the constant c [Ans c=1/2 ]
1
ii) p(x ≤ ) [Ans 1-(1/2)1/2 ]
2

54) A family of pdf’s that has been used to approximate the distribution of
income, city population size, and size of firms is the Pareto family. The
family has two parameters, k and θ, both > 0, and the pdf is
𝑘 ∙ 𝜃𝑘
ƒ(x;, k, θ) ={ 𝑥 𝑘+1 𝑥 ≥ 𝜃
0 𝑥< 𝜃
60 | P a g e RAHUL STUDY GROUP

a) Verify that the total area under the graph equals 1.

b) If the rv X has pdf ƒ(x; k, θ), for any fixed b > θ, obtain an
𝜽 𝒌
expression for P(X ≤ b). [Ans 1 –( ) ]
𝒃
c) For θ < a< b, obtain an expression for the probability P(a ≤ X ≤
𝜽 𝜽 𝒌
b). [Ans ( ) 𝒌 − ( ) ]
𝒂 𝒃

Expectation And Variance, Median

55) Suppose that the pdf of the random variable x is as follows:

1
𝑥 ,0 ≤ 𝑥 ≤ 4
f(x)={8
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Find E(X) and V(X) [Ans 8/3 ,8/9]

56) find the E(X) and V(X) of the random variable X whose probability
density is given by.
𝑥 𝑓𝑜𝑟 0 < 𝑥 < 1
𝑓(𝑥) = {2 − 𝑥 𝑓𝑜𝑟 1 ≤ 𝑥 < 2}
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

57) An ecologist wishes to mark off a circular sampling region having

radius 10 m. However, the radius of the resulting region is actually a
random variable R with pdg
3
[1 − (10 − 𝑟)2 ] 9 ≤ 𝑟 ≤ 11
ƒ(r) = {4
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
what is the expected area of the resulting circular region? [Ans
100.2𝝅]

𝑘.θ𝑘
58) 𝑓(𝑥; 𝑘) = {𝑥 𝑘 +1 𝑥 ≥ θ}
0 𝑥<𝜃
𝒌𝜽
a) If k > 1, compute E(x) [Ans ]
𝒌−𝟏
b) What can you say about E(x) if k = 1 [∞]
c) If k > 2 show that v(x) = kθ2 (𝑘 − 1)−2 (𝑘 − 2)−1
d) If k=2 what is v(x) [Ans ∞]
61 | P a g e RAHUL STUDY GROUP

e) What conditions on k are necessary to ensure that E(𝑥 𝑛 ) is infinity?

0 𝑥 < −2
1 3 𝑥3
59) 𝐹(𝑥) = { + (4𝑥 − ) − 2 ≤ 𝑥 < 2}
2 32 3
1 2≤𝑥
a) P(x < 0) [Ans 0.5]
b) P(-1 < x < 1) [Ans 0.6875]
c) P(x > 0.5) [Ans 0.3164]
d) Find PDF [Ans 0.09375( 4-x2)]
e) Verify 𝑢̃=0

60) Let X denote the amount of time a book on two- hour reserve is actually
checked out, and suppose the cdf is
0 𝑥 <0
𝑥 2
F(x)={ 0 ≤ 𝑥 < 2
4
1 2 ≤𝑥
Use the cdf to obtain the following:
a) P(X ≤ 1) [Ans. 0.25]
b) P(.5 ≤ X ≤ 1) [Ans. 0.1875]
c) P(X > 1.5) [Ans 0.9375]
d) The median checkout duration ũ? [Ans 1.414]
𝒙
e) F’ (x) to obtain the density function ƒ(x) [Ans ]
𝟐
f) E(X) [Ans 1.333]
g) V(X) and 𝜎𝑥 ? [Ans .222 , .471]
h) If the borrower is charged an amount h(X) = 𝑋 2 when checked
duration is X, compute the expected charge E[h(X)]. [Ans 2]

0 𝑥≤0
𝑥 4
61) 𝐹(𝑥) = { [1 + 1𝑛 ( )] 0 < 𝑥 ≤ 4}
4 𝑥
1 𝑥>4
a) P(𝑥 ≤ 1) [Ans 0.597]
b) P(1 ≤ 𝑥 ≤ 3) [Ans 0.369]
c) PDF [Ans .25In(4) -.25In(x)]
d) Find E(X) ,V(X) ,Median
62 | P a g e RAHUL STUDY GROUP

62) If the probability density function is given by

𝑥
,0 < 𝑥 ≤ 1
2
1
,1 < 𝑥 ≤ 2
f(x)= 2
3−𝑥
,2 < 𝑥 < 3
2
{ 0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Find the expected value of g(x)=𝑥 2 − 5𝑥 + 3? [Ans -11/6 ]

63) The weekly demand for propane gas ( in 1000s of gallons) from a
particular facility is an rv X with pdf

1
2(1 − 2 ) , 1 ≤ 𝑥 ≤ 2
f(x)={ 𝑥
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

a. Compute the cdf of X

b. Obtain the expression of 100the percentile ,what is the value of 𝑢̃
c. Compute E(X) and V(X)
d. If 1.5 thousand gallons are in stock at the beginning of the week
,how much of the 1.5 thousand gallons is expected to be left at the
end of the week ?

64) Suppose that in a different traffic environment ,the distribution of a

time headway has the form

𝑘
, 𝑥>1
f(𝑥)={𝑥 4
0 ,𝑥 ≤ 1

a. Determine the value of k for which f(𝑥) is a legitimate pdf.

b. Obtain the cumulative distribution function
63 | P a g e RAHUL STUDY GROUP

c. Use the cdf form(b) to determine the probability that headway is

between 2 and 3 sec.
d. Obtain the mean value of headway and the standard deviation of the
headway.
e. What is the probability that the headway is within 1 Standard
deviation of the mean value ?

Miscellaneous
1) The probability density of a random variable is 𝑓(𝑥) = 𝑎𝑥 2 𝑒𝑥𝑝−𝑘𝑥 (𝑘 >
0,0 ≤ 𝑥 ≤ ∞)
𝒌𝟑
Then, a equals [Ans ]
𝟐

2) A continuous random variable X has a probability density function f(x)

= 3x2
𝟏
𝟏 𝟑
with 0 ≤ x ≤ 1. If P(X ≤ a) = P(x >a), then a is: [Ans (𝟐) ]

3) If a probability density function of a random variable X is given by

𝑓(𝑥) = 𝑘𝑥(2 − 𝑥), 0 ≤ 𝑥 ≤ 2, then mean of X is [Ans 1]

4) A fair coin is tossed repeatedly until a head is obtained for the first time.
Let X denotes the number of tosses that are required. The value of the
distribution function of X at 3 is [Ans 7/8]

5) Suppose θ is a random variable with uniform distribution on the interval

[-π/2, π/2]. The value of the distribution function of the random
variable X = sin θ at x∈ [-1,1] is [Ans 𝐬𝐢𝐧−𝟏 (𝐱) /𝛑 + 𝟏/𝟐]

6) The next three questions are based on the following data. The number of
loaves of bread sold by a bakery in a day is a random variable X. The
distribution of X has a probability density function f is given by
64 | P a g e RAHUL STUDY GROUP

k𝑥 if x ∈ [0, 5)
f(𝑥) = {k(10 − 𝑥) if x ∈ [5, 10) }
0, if x ∈ [10, ∞)

a) As f is a probability density function, the value of k ? [Ans 1/25]

b) Let A be the event that X ≥ 5 and let B be the event that X ∈ [3, 8].
The probability of A conditional on B ? [Ans 21/37 ]

c) Events A and B are Independent ? [Ans Yes]

7) The continuous random variable X has probability density f(𝑥) where

a if 0 ≤ 𝑥 < 𝑘
f(𝑥) = {b if k ≤ 𝑥 ≤ 1 }
0 otherwise
𝟏−𝟐𝐛+𝐚𝐛
where a>b>0 and 0<k<1 .Then E(X) is given by [ ]
𝟐(𝐚−𝐛)

8) An unbiased coin is tossed until a head appears. The expected number of

tosses required ? [Ans 2 ]

9) A continuous random variable x has the following probability density

function:
α x0 α+1
f(x) = ( ) for x > x0 , α > 1
x0 x

The distribution function and the mean of x are given respectively

𝐱 −𝛂 𝛂𝐱 𝟎
by[Ans 𝟏 − ( ) , ]
𝐱 𝟎 𝛂−𝟏

10) A machine starts operating at time 0 and fails at a random time T.

t
1 −
The distribution of T has density f(t) = 𝑒 3 for t > 0. The machine will
3
not be monitored until time t = 2. The expected discovery time of the
𝟐
−
machine’s failure ? [Ans 2+3𝒆 ] 𝟑
65 | P a g e RAHUL STUDY GROUP

11) An insuree has an insurance policy against a random loss X ∈ [0, 1]. If
loss X occurs then the insurer pays X − C to the insuree, who bears the
remaining loss C ∈ (0, 1). The loss X is a continuous random variable
with density function f(x) = 2x, 0 < x < 1 . If the probability of insurance
payment being less than 1/2 is 0.64, then C ? [Ans 0.3]

12) Let Y denote the number of heads obtained when 3 fair coins are
tossed. Then, the expectation and Variance of Z = 4 + 5𝑌 2
[Ans 19 ,187.5 ]
13) An urn contains equal number of green and red balls. Suppose you
are playing the following game. You draw one ball at random from the
urn and note its color . The ball is then placed back in the urn, and the
selection process is repeated. Each time a green ball is picked you get 1
Rupee. The first time you pick a red ball, you pay 1 Rupee and the game
ends. Your expected income from this game is [Ans 0 ]

14) Let X denotes the absolute value of difference between the number
obtained when two dice are tossed . the expectation of X. [Ans 35/18]

15) On this tour, a night watchman has to open a door in the dark. He has
20 keys. Only one of which fits the lock. He makes use of two different
methods to open the door Method A: He carefully tries the keys one by
one to avoid using the same key twice. Method B: He tries the keys at
random. Define the random variables XA and XB as the number of
necessary trials to open the door when using the method A and B
respectively. Work out the probability distribution of XA and XB.

16) Show that the CDF F(x) is a non-decreasing function;

that is, 𝑥1 < 𝑥2 implies that F(𝑥1 ) ≤ F(𝑥2 ). Under what conditions
F(𝑥1 ) = F(𝑥2 )? (When P(𝒙𝟏 < 𝑿 ≤ 𝒙𝟐 ) = 0)

17) If a ≤ 𝑋 ≤ b ,Show that 𝑎 ≤E(X) ≤ 𝑏

66 | P a g e RAHUL STUDY GROUP

CHAPTER -5

Probability Distribution

BERNOULLI DISTRIBUTION

1. Let X be a Bernoulli rv with pmf.

a. Compute E(X2) (Ans:p)

b. Show that V(X) = p(1 – p)

c. Compute E(X19) (Ans:p)

2. Suppose that X is a random variable such that E(𝑥 𝑘 )=1/3 for k = 1, 2, ….. . assuming
that there can’t be more than one distribution with the same sequence of moments,
𝑃(𝑥 = 0) = 2/3
determine the distribution of x? {
𝑃(𝑥 = 1) = 1/3

BINOMIAL DISTRIBUTION

1. Compute the following binomial probabilities directly from the formula for
b(𝑥 ; , 𝑛, 𝑝) ∶
a. b (3; 8 , .6 ) [Ans 0.124 ]
b. b (5; 8 , .6 ) [Ans 0.279 ]
c. P ( 3≤ 𝑋 ≤ 5) when n =8 and p= 0.6 [Ans 0.635 ]
d. P(X≥ 1) when n =12 and p=0.1 [Ans 0.718 ]

2. Use the Binomial CDF table , determine the following probabilities

a. B(4 : ,10 , .3 ) [Ans 0.850 ]

b. b(6 : ,10 , .7 ) [0.200 ]
c. P(2≤ 𝑋 ≤ 6) when X ~ 𝐵𝑖𝑛(15, 0.3) [0.8336 ]
67 | P a g e RAHUL STUDY GROUP

3. A particular telephone number is used to receive both voice call and fax message
. Suppose that 25% of the incoming calls involve a fax messanger and consider a
sample of 20 incoming calls. What is the probability that
a. At most 6 of the calls involve a fax message ? [ Ans 0.786 ]
b. Exactly 6 of the calls involve a fax message ? [Ans 0.169 ]
c. At least 6 of the calls involve a fax message ? [0.383 ]

4. When circuit boards used in the manufacture of compact disc player are tested,
the long run % of defectives is 30%. Let X= the number of defective boards in a
random sample of size n = 15 , so X ~ 𝐵𝑖𝑛(15, 0.3)

a) Determine P(2 < X < 6)

b) Determine P(X ≥ 2)
c) Determine P(2≤ 𝑋 ≤ 4)
d) Calculate the expected value and variance of X?

5. A company that produces fine crystals knows from experience that 10% of its
goblets have cosmetic flaws and must be classified as “seconds.”
a. Among 6 randomly selected goblets, how likely is it that only one is selected?
(0.354)
b. Among 6 randomly selected goblets, what is the probability that at least two
are seconds? (0.115)
c. If goblets are examined one by one, what is the probability that at most 5
must be selected to find four that are not seconds? (0.918)

6. Suppose that the probability that a certain experiment will be successful is 0.4
and let X denote the number of success that are obtained in 15 independent
performance of an experiment. Find the probability p(6 ≤ 𝑋 ≤ 9)? (0.5630)

7. The probability that a student will graduate is 0.4. Determine the probability
that out 5 students:
(i) none; (ii) 1; (iii) at least 1; (iv) all, will graduate [Ans.: (i)
0.07776, (ii) 0.2592, (iii) 0.92224, (iv) 0.01024]

8. A coin is tossed 9 times for which the probability of head is 0.6. Find the
probability of obtaining an even number of heads? (0.5000)

9. 20% of all telephones of a certain type are submitted for service while under
warranty. Of these, 60% can be repaired, whereas the other 40% must be
replaced with new units. If a company purchases 10 of these telephones, what is
the probability that exactly two will end up being replaced under warranty?
(0.1478)
68 | P a g e RAHUL STUDY GROUP

10. Suppose that 90% of all batteries from a certain supplier have acceptable
voltages. A certain type of supplier has acceptable voltages. A certain type of
flashlight requires two type-D batteries, and the flashlight will work only if both
its batteries have acceptable voltages. Among 10 randomly selected flashlights,
what is the probability that at least nine will work, given that the events are
independent? (0.407).

11. Find the binomial distribution if its mean is 48 and the standard deviation is 4.

12. Obtain the mean and Standard deviation of a binomial distribution for which P(X
= 3) = 16P(X = 7) and n = 10. [Ans:3.33 ,1.49]

13. The mean of a binomial distribution is 20 and standard deviation is 4. Calculate

n, p and q with usual notations. [Ans.: n = 100, P = 1/5, q = 4/5]

14. In a binomial distribution with 6 independent trials, the probability of 3 and 4

successes is found to be 0.2457 and 0.0819 respectively. Find the parameters p
and q of
the binomial distribution. [[Ans.: 4/13, 9/13]

15. An experiment succeeds twice as often as it fails. What is the probability that in
next five trials there will be (i) three successes, (ii) at least three successes?
[Ans.: (i) 80/243, (ii) 64/81]

16. . The probability of a man hitting the target is ½. How many times .must he fire
so that the probability of his hitting the target at least once is greater than 90%?
[Ans.: At least 4 times]

17. An insurance salesman sells policies to 5 men all of identical age and good health.
According to the actuarial tables, the probability that a man of this particular age
will be alive 30 years hence is 2/3. Find the probability that 30 years hence
i) At least 1 man will be alive,
ii) At least 3 men will be alive.
[Ans. : (i)1 −
1 5 64
(3) , (ii) 81]
69 | P a g e RAHUL STUDY GROUP

18. Let the random variable X follow a Binomial distribution with parameter n and p
where n(>1) is an integer and 0 < p < 1. Suppose further that the possibility of X
= 0 is the same as the probability of X = 1. Then the value of p is? [1/(n+1)]

19. Experience shows that 20% of the people reserving tables at a certain restaurant
never show up. If the restaurant has 50 tables and takes 52 reservations, then
4 52
the probability that it will be able to accommodate everyone? (𝟏 − 𝟏𝟒 × (5) )

20. The College Board reports that 2% of the two million high school students who
take the - each year receive special accommodations because of documented
disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25
students who have recently taken the test.

a.What is the probability that exactly 1 received a special accommodation? [Ans

0.308 ]

b.What is the probability that at least 1 received a special accommodation? [Ans

0.397 ]

c.What is the probability that at least 2 received a special accommodation? [Ans

0.295 ]

d.What is the probability that the number among the 25 who received a special
accommodation is within 2 standard deviations of the number you would expect
to be accommodated?[Ans .705 ]

e.Suppose that a student who does not receive a special accommodation is

allowed 3 h for the exam, whereas an accommodated student is allowed 4.5 h.
What would you expect the average time allowed the 25 selected students to be?
[Ans 3.03 ]

21. Prove the below question

a. show that b(x; n, 1-p)=b(n-x; n, p)

b. show that B(x; n, 1-p)=1-B(n-x-1; n, p)

22. Out of 1000 families of 3 children each, how many families would you expect to
have two boys and 1 girls assuming that boys and girls are equally likely? (375)

23. Answer the below question

a. For fixed value of n are there value p for which V(X) =0 ? Explain why this
so
70 | P a g e RAHUL STUDY GROUP

b. For what value of p V(X ) is maximum ?

Hypergeometric Disrtibution

1. Suppose that five red and 10 blue balls . if seven balls are selected at random
without replacement ,what is the probability that at least three red balls will be
obtained ? [Ans 0.4266 ]

2. A group of 6 female managers and 19 male managers apply for an assignement . A

random sample of 5 people is drawn from a hat (without replacement ) . What is
the probability 4 of the chosen mangers will be female and 1 will be male ?[
0.0053642 ]

3. Find the Expectation of a Hypergeometric Distribution such that the probability

that a 4-trial hypergeometric experiment results in exactly 2 successes, when the
population consists of 16 items. [Ans ½ ]

4. If a random variable X has the hypergeometric distribution with parameter M=8

,N=28 and n ,for what value of n will V(X) is maximum ? [Ans 14 ]

5. An electronics store has received a shipment of 15 table radios that have

connections for an iPod or iPhone. Six of these have two slots (so they can
accommodate both devices), and the other eight have a single slot. Suppose that
Five of the 15 radios are randomly selected to be stored under a shelf where the
radios are displayed, and the remaining ones are placed in a storeroom. Let X=the
number among the radios stored under the display shelf that have two slots.
a) What kind of a distribution does X have (name and values of all parameters)?
[Ans Hypergoemtric ]
b) Compute P(X=2),P(X≤ 2), and P(X ≥ 2). [Ans 0.280 ,0.573 .706]
c) Calculate the mean value and standard deviation of X. [Ans .857 ,.926 ]

6. Each of 12 refrigerators of a certain type has been returned to a distributor because

of an audible, high-pitched, oscillating noise when the refrigerators are running.
Suppose that 7 of these refrigerators have a defective compressor and the other 5
have less serious problems. If the refrigerators are examined in random order, let X
be the number among the first 6 examined that have a defective compressor.
Compute the following:
a) P(X =5) [Ans 0.114 ]
b) P(X≤ 4) [Ans 0.879 ]
c) The probability that X exceeds its mean value by more than 1 standard deviation.
[Ans 0.121 ]
71 | P a g e RAHUL STUDY GROUP

d) Consider a large shipment of 400 refrigerators, of which 40 have defective

compressors. If X is the number among 15 randomly selected refrigerators that
have defective compressors, describe a less tedious way to calculate (at least
approximately) P(X≤ 5) than to use the hypergeometric pmf. [ Ans 0.998 ]

7. An instructor who taught two sections of engineering statistics last term, the first
with 20 students and the second with 30, decided to assign a term project. After all
projects had been turned in, the instructor randomly ordered them before grading.
Consider the first 15 graded projects.
a) What is the probability that exactly 10 of these are from the second section? [Ans
0.2070 ]
b) What is the probability that at least 10 of these are from the second section? [Ans
0.3799 ]
c) What is the probability that at least 10 of these are from the same section? [Ans
0.39402 ]
d) What are the mean value and standard deviation of the number among these 15
that are from the second section?
e) What are the mean value and standard deviation of the number of projects not
among these first 15 that are from the second section?

8. A personnel director interviewing 11 senior engineers for four job openings has
scheduled six interviews for the first day and five for the second day of
interviewing. Assume that the candidates are interviewed in random order.
a) What is the probability that x of the top four candidates are interviewed on the
first day? [Ans Pdf ]
b) How many of the top four candidates can be expected to be interviewed on the
first day? [Ans 2.18 ]

9. Twenty pairs of individuals playing in a bridge tournament have been seeded 1, . . . ,

20. In the first part of the tournament, the 20 are randomly divided into 10 east–
west pairs and 10 north–south pairs.
a) What is the probability that x of the top 10 pairs end up playing
east–west?
b) What is the probability that all of the top five pairs end up playing the same
direction? [Ans 0.033 ]
c) If there are 2n pairs, what is the pmf of among the top n pairs who end up
playing east–west? What are E(X) and V(X)? [Ans n/2 ,n/2n-1 .n/4]

10. A second-stage smog alert has been called in a certain area of Los Angeles
County in which there are 50 industrial firms. An inspector will visit 10
randomly selected firms to check for violations of regulations.
a) If 15 of the firms are actually violating at least one regulation, what is the pmf
of the number of firms visited by the inspector that are in violation of at least
one regulation? [Ans PDF ]
72 | P a g e RAHUL STUDY GROUP

b) If there are 500 firms in the area, of which 150 are in violation, approximate
the pmf of part (a) by a simpler pmf. [Ans Binomial ]
c) For X=the number among the 10 visited that are in violation, compute E(X)
and V(X) both for the exact pmf and the approximating pmf in part (b). [ Ans
3 ,2.06 ,3 ,2.1 ]
11. A geologist has collected 10 specimens of basaltic rock and 10 specimens of
granite. The geologist instructs a laboratory assistant randomly select 15 of the
specimens for analysis.
(a) What is the pmf of the number of granite specimens selected for analysis?
(b) What is the probability that all specimens of one of the two types of rock are
selected for analysis? [Ans 0.0326 ]
(c) Calculate the mean and standard deviation (up to four decimal places) for X, the
number of granite specimens selected. [Ans 7.5 , 0.9934 ]
(d) Find the probability that the number of granite specimens selected for analysis is
within 1 standard deviation of its mean value, that is, find P(μ−σ≤X≤μ+σ). [Ans
.6966]

12. Suppose that in a large lot containing T , manufactured items , 30 percent of the
items are defective and 70 percent are non defective . Also suppose that ten items
are selected at random without replacement from the lot . Determine

a. An exact expression for the probability that not more than one defective
item will be obtained ?
b. An approximation for this probability based on the binomial distribution ?

Geometric Distribution

1. If the Probability of having a male or female child is equal ,What is the

probability that a women fourth Child is her first son? [Ans :.0625 ]

2. An oil company conducts a geological study that indicates that an exploratory oil
well should have a 20% chance of striking oil. What is the probability that the
first strike comes on the third well drilled? [Ans 0.128 ]

3. Bob is a high school basketball player who has a 70% free throw percentage.
Assume all free throw attempts are independent of one another.

a. What is the probability his first made free throw is on the third shot? [Ans 0.063 ]
73 | P a g e RAHUL STUDY GROUP

b. What is the probability it takes more than 3 shots to get his first made free throw?
[Ans .027 ]

c. Probability that first success on xth trial lying within 1 S.D of mean value ? [Ans 0.91 ]

NEGATIVE BINOMIAL DISTRIBUTION

1. If the Probability that a person will believe a rumour about a scandal in politics
is 0.8, Find the probability that the ninth person to hear the rumour will be the
fourth person to believe it. [Ans: .00734]

2. Bob is a high school basketball player who has a 70% free throw percentage.
Assume all free throw attempts are independent of one another.
c. What is the probability that his third made free throw is on his fifth shot?
[Ans 0.1852 ]
d. What is the probability that his 100th made free throw is on his 123rd
shot? [Ans 0.0012 ]

3. The Minnesota Twins are having a bad year. Suppose their ability to win any one
game is 42% and games are independent of one another.
a. What is the probability it takes 14 games for them to win their fourth game?
[Ans 0.0383 ]
b. What is the expected value and variance of the number of games it will take
them to win their fortieth game? [Ans 95.3821 ,131.5193 ]

4. A family decides to have children until it has three children of the same gender.
Assuming P(B)=P(G)=0.5, what is the pmf of X=the number of children in the
family?

5. A footballer’s probability of scoring a goal on one attempt is 0.8. The player’s

coach wishes to determine the probability that the player’s 3rd goal would be on
his 5th attempt. [Ans .12288 ]

6. You roll a die until you get four sixes (not necessarily consecutive). What is the
mean and standard deviation of the number of rolls you will make? [Ans 24
,10.95 ]

7. Suppose that sequence of independent tosses are made with a coin for which the
probability of obtaining a head on each given toss is 1/30.

a. What is probability of obtaining 3 tails before 5 heads have been obtained .

b. What is the expected number of tails that will be obtained before five heads have
been obtained ? [Ans 145 ]
74 | P a g e RAHUL STUDY GROUP

c. What is the variance of the number of tails that will be obtained before five heads
have been obtained ? [Ans 4350 ]

8. P(Success )= 0.8 , P(Failure) =0.2.

a. Find the probability of obtaining exactly seven failure before the third
success ? [Ans .0001207]
b. Find the probability of obtaining at most 3 failure before the fifth success ?
[Ans .6160384]

9. Suppose that two players A and B are trying to throw a basketball through a
hoop. The probability that player A will succeed on any given throw is p, and he
throws until he has succeeded r times. The probability that player B will succeed
on any given throw is mp, where m is a given integer
(m = 2, 3, . . .) such that mp < 1, and she throws until she has succeeded mr times.

a) For which player is the expected number of throws smaller? [Same]

b)
c) For which player is the variance of the number of throws smaller? [Player B ]

10. Suppose that p=P(male birth)=0.5. A couple wishes to have exactly two female
children in their family. They will have children until this condition is fulfilled.
a) What is the probability that the family has x male children?
b) What is the probability that the family has four children? [Ans 0.188 ]
c) What is the probability that the family has at most four children? [Ans 0.688 ]
d) How many male children would you expect this family to have? How many
children would you expect this family to have? [Ans 2 ,4 ]

UNIFORM DISTRIBUTION

1. If X follows Uniform Distribution with interval [a,b]..Find out their Mean And
variance

2. Observe depth 𝑋~ ∪ (7.5,20)

a) Find the mean and variance.

b) What is the probability that observed 𝑥 ≤ 10.
c) What is the prob. that observed depth is within 1 standard deviation of the mean
value? Within 2 standard deviation.
75 | P a g e RAHUL STUDY GROUP

3. In certain experiments, the error made in determining the density of a substance

is a random variable having a uniform density with 𝛼 = −0.015 and 𝛽 = 0.015.
Find the probabilities that such an error will

a) Be in b/w -0.002 and 0.003? (1/6)

b) Exceed 0.005 in absolute value? (2/3 )

4. Let X have the uniform distribution on the interval [a, b] and c is greater than 0. Prove
that cx+d has the uniform distribution on the interval [ca+d, cb+d]?

5.. Suppose that in a quiz there are 30 participants. A question is given to all participants
and the time allowed to answer it is 25 seconds. Find the probability of participant
respond within 6 seconds? (6/25)

6. Suppose that a random variable N is taken from 690 to 850 minutes in uniform
distribution. Find the probability that N is greater than 790? (0.1)

7. The time in minutes that A takes to checkout at her local supermarket follows a
continuous uniform distribution over the interval [3, 9].

a) A’s expected checkout time. (6)

b) The variance of the time taken to checkout at the supermarket. (3)
c) The probability that A will take more than 7 minutes to checkout. (1/3)

Given that A has already spent 4 minutes at the checkout.

d) Find the probability that she will take a total of less than 6 minutes to
checkout. (2/5)

8. In a game, player select sticks at random from a box containing a large number of
sticks of different lengths. The length of a randomly chosen stick has a continuous
uniform distribution over the interval [7, 10].

a) A stick is selected at random from the box, what is the probability that the
stick is shorter than 9.5 cm. (5/6)
b) To win a bag of sweets, a player must select 3 sticks and wins if the length of
the longest stick is more than 9.5cm. What is the probability of winning a bag
of sweets.
c) To win a soft toy, a player must select 6 sticks and wins the toy if more than 4
of the sticks are shorter than 7.6cm. What is the probability of winning a soft
toy.
76 | P a g e RAHUL STUDY GROUP

9. Suppose that you are conducting a quiz and post a question to the audience of 20
competitors. The time allowed to answer the question is 30 second. How many person
are likely to respond within 5 seconds? (3)

10. At a certain station a train arrives every 2 minutes if a passenger reaches the platform
when there is no waiting train, what is the probability that

a) He will have to wait for more than 45 second for the next train.
b) What is the variance of the waiting time. [Ans:.625, 1200]

11. Suppose that the random values 𝑋1,…., 𝑋𝑛 𝑓𝑜𝑟𝑚 a random sample of size n from the
uniform distribution on the interval [0, 1]. Let 𝑌1 =min.{𝑋1 , … . . , 𝑋𝑛 }, and let 𝑌𝑛 =
𝟏 𝒏
𝑚𝑎𝑥. {𝑋, … . . , 𝑋𝑛 }. Find the E(𝑌1 ) and E(𝑌𝑛 )? E(𝒀𝟏 ) = 𝒏+𝟏 , 𝑬(𝒚𝒏 ) = 𝒏+𝟏

EXPONENTIAL DISTRIBUTION

1. Let X denote the time b/w two successive arrivals at the drive up window of a
local bank. If X has an exponential distribution with ƛ=1, compute the
following:
a) The expected time between two arrivals (1)
b) The SD of the time b/w successive arrivals (1)
c) P(X ≤ 4) (0.982)
d) P(2≤ 𝑋 ≤ 5) (0.129)

2. Data collected at IGI AIRPORT suggests that an exponential distribution with

mean value 2.725 hours is good model for rainfall duration.
a) What is the probability that the duration of a particular rainfall event at
this location is at least 2 hours? (0.480)
b) At most 3 hours? (0.667)
c) B/w 2 and 3 hours? (0.147)
d) What is the probability that rainfall duration exceeds the mean value by
more than 2 sds? (0.050)
e) What is the probability that the rainfall duration is less than the mean
value by more than 1 sd? (0)

3. Suppose that the time one spends in bank is exponentially distributed with
mean = 10 minutes.
77 | P a g e RAHUL STUDY GROUP

a) What is the probability that customer will spend more than 15 minutes in
bank? (0.22)
b) What is the probability that a customer will spend more than 15 minutes
in bank given that he is still in the bank after 10 minutes? (0.604)

4. If jobs arrive every 15 seconds on average, ƛ=4 per minute, what is the
probability of waiting less than or equal to 30 seconds? (0.86)

5. Calls arrive at an average rate of 12 per hour. Find the probability that a call
will occur in the next 5 minutes given that you have already waited 10
minutes for a call?
(0.63)

6. The lifetime T (years) of an electronic component is a continuous random

variable with a pdf given by

f(t)=𝑒 −𝑡 , 𝑡 ≥ 0

Find the lifetime L which a typical component is 60% certain to exceed. If 5

Components are sold to a manufacturer, find the probability that atleast
One of them will have a lifetime less than L years? (0.92)

7. If X has an exponential distribution with parameter ƛ, what is the probability

that a random variable X is less than its expected value? (1-exp(-1))

8. Let X be an exponential random variable with parameter ƛ=ln(3).Compute

the probability p (2≤ 𝑋 ≤ 3)? (3−2 − 3−4 )

NORMAL DISTRIBUTION

1. Let Z be a standard normal random variable and calculate the

following probabilities, drawing pictures wherever appropriate.
a) P(0≤ Z ≤ 2.17) [Ans 0.4850 ]
b) P(0≤ Z ≤ 1) [Ans 0.3413 ]
c) P(-2.50≤ Z ≤ 0) [Ans 0.4938]
d) P(-2.50≤ Z ≤ 2.50) [Ans .9876 ]
78 | P a g e RAHUL STUDY GROUP

e) P(Z ≤ 1.37) [Ans .9147 ]

f) P(-1.75≤ Z) [Ans 0.9599 ]
g) P(-1.50≤ Z ≤ 2.00) [Ans .9104 ]
h) P(1.37≤ Z ≤ 2.50) [Ans .0791 ]
i) P(1.50≤ Z) [Ans .0668 ]
j) P(|Z| ≤ 2.50) [Ans 0.9876]

2. In each case, determine the value of the constant c that makes the
probability statement correct.
a) Φ(c) = .9838 [Ans 2.14 ]
b) P(0≤ Z ≤ c)=.291 [Ans 0.81 ]
c) P(c ≤ Z)=.121 [Ans 1.17 ]
d) P(-c≤ Z ≤ c)=.668 [Ans 0.97]
e) P(c≤ |Z|)=.016 v [Ans 2.41 ]

3. Find the following percentiles for the standard normal distribution.

Interpolate where appropriate.
a) 91st [Ans 1.34 ]
b) 9th [Ans -1.34 ]
c) 75th [Ans 0.675 ]
d) 25th [Ans -0.675 ]
e) 6th [Ans -1.555 ]

4. The P.D.F of a normal distribution is

−(𝑥−5)2
f(x)=k.exp[ 18
]

1
a) what is the value of k? k=(3.√2𝜋)
b) what is the mean value of the distribution? Mean=(5)
c) What is the standard deviation of the distribution? S.d=(3)

5. A project yields an average cash-flow of Rs. 550 lakhs and standard deviation
cash-flow of Rs. 110 lakhs. Calculate the following probabilities assuming the
normal distribution:
(i) Cash flow will be more than Rs. 750 lakhs
(ii) Cash flow will less than RS. 450 lakhs;
(iii) Cash flow will be between Rs. 425 lakhs and Rs. 750 lakhs.

[Ans.: (i) 0.1271, (ii) 0.1814, (iii) 0.8385]

79 | P a g e RAHUL STUDY GROUP

6. The mean and standard deviation of the wages of 6,000 workers engaged in a factory
are Rs. 1,200 and Rs. 400 respectively. Assuming the distribution to be normal estimate:

i) Percentage of workers getting wages above Rs. 1,600,

ii) Number of workers getting wages between Rs. 600 and Rs. 900,

iii) Number of workers getting wages between Rs. 1,100 and Rs. 1,500.

[Ans.: i) 15.87, ii) 959, iii) 2233]

7. The weekly wages of 2000 workers are normally distributed. Its mean and standard
deviation are Rs. 140 and Rs. 10 respectively. Estimate the number of workers whose
weekly wages will be:

i) between Rs. 120 and Rs. 130

ii) more than Rs. 170

iii) less than Rs. 165

[Ans.: (i) 272, (ii) 3, (iii) 1988]

8. There are two machines available for cutting corks intended for use
in wine bottles. The first produces corks with diameters that are
normally distributed with mean 3 cm and standard deviation .1 cm.
The second machine produces corks with diameters that have a
normal distribution with mean 3.04 cm and standard deviation .02
cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm.
Which machine is more likely to produce an acceptable cork?

9. A machine that produces ball bearings has initially been set so that
the true average diameter of the bearings it produces is .500 in. A
bearing is acceptable if its diameter is within .004 in. of this target
value. Suppose, however, that the setting has changed during the
course of production, so that the bearings have normally distributed
diameters with
mean value .499 in. and standard deviation .002 in. What percentage of
the bearings produced will not be acceptable? [Ans 0.073]

10. The income distribution of officers of a certain company was found to follow
normal distribution. The average income of an officer was Rs, 15,000. The
standard deviation of the income of officers was Rs, 5,000. If there: were 242
80 | P a g e RAHUL STUDY GROUP

officers drawing salary above Rs. 18,500, how many officers were there in the
company?

[Ans.: 1000]

11. The incomes of a group of 10,000 persons were found to be normally distributed
with mean equal to Rs. 750 and standard deviation equal to Rs. 50. What was
lowest income among the richest 250?

[Ans.: Rs. 848]

12. The daily wages of 1,000 workmen are normally distributed around a mean of Rs.
70 and with a standard deviation of Rs. 5. Estimate the number of workers whose daily
wages will be:

i) Between Rs. 70 and 72

ii) Between Rs. 69 and 72

iii) More than Rs. 75

iv) Less than Rs. 63

v) More than Rs. 80

Also estimate the lowest daily wages of the 100 highest paid workers.
[Ans.: i) 155, ii) 235, iii) 159, iv) 81, V) 23 and Lowest daily wages = Rs. 76.40]

13. The marks of the students in a certain examination are normally distributed
with mean marks as 40% and standard deviation marks as 20%. On this basis,
60% students failed. The result- was moderated and 70% students passed. Find
the pass marks before and after the moderation.

[Ans.: 45%, 29.6%]

14. A set of examination marks is approximately normally distributed with a mean

of 75 and standard deviation of 5. If the top 5% of students get grade A and the
bottom 25% get grade D, what is the lowest A and what marks are the highest D.
[Ans.: 83.225; 71.65]

15. Suppose that blood chloride concentration (mmol/L) has a

normal distribution with mean 104 and standard deviation 5
Assuming Normal Distribution
a) What is the probability that chloride concentration equals 105? Is
less than 105? Is at most 105? [Ans 0 ,0.5793 ]
81 | P a g e RAHUL STUDY GROUP

b) What is the probability that chloride concentration differs from the

mean by more than 1 standard deviation? Does this probability
depend on the values of μ and σ? [Ans .3174 ,No ]
c) How would you characterize the most extreme .1% of chloride
concentration values? [Ans 87.6 ,120.4]

16. Yield strength for grade steel is normally distributed with μ =

43 and σ = 4.5. What yield strength value separates the strongest
75% from the others? [Ans 39.985 ]

17. The temperature reading from a thermocouple placed in a

constant-temperature medium is normally distributed with mean μ,
the actual temperature of the medium, and standard deviation σ.
What would the value of σ have to be to ensure that 95% of all
readings are within .10 of μ? [Ans 0.0510 ]

18. The average weekly food expenditure of families in a certain area has a normal
distribution with mean Rs. 125 and standard deviation Rs. 25. What is the
probability that a family selected at random from this area will have an average
weekly expenditure on food in excess of Rs. 175? What is the probability that out
of eight such families selected at least one family will have their weekly food
expenditure in excess of Rs. 175?

[Ans.: 0.0228, 0.1685]

19. The automatic opening device of a military cargo parachute has

been designed to open when the parachute is 200 m above the
ground. Suppose opening altitude actually has a normal distribution
with mean value 200 m and standard deviation 30 m. Equipment
damage will occur if the parachute opens at an altitude of less than
100 m. What is the probability that there is equipment damage to the
payload of at least one of five independently dropped parachutes?
[Ans 0.0004 , 0.002 ]

20. In a distribution exactly normal, 7% of the items are under 35 and 89% are
under 63. What are the Mean and Standard Deviation of the distribution?

[Ans.: 50.29 & 10.33]

82 | P a g e RAHUL STUDY GROUP

21. The marks of the students are normally distributed. 10% get more than 75
marks and 20% get less than 40 marks. Find the mean and standard deviation of
the distribution

[Ans.: 53.87; 16.51]

22. Time taken by a construction company to construct a flyover is a normal variate

with mean 400 labour days and standard deviation of 100 labour days. If the company
promises to construct the flyover in 450 days or less and agrees to pay a penalty of Rs.
10,000 for each labour day spent in excess of 450, what is the probability that:

i) The company pays a penalty of at least Rs. 2, 00,000?

ii) The company takes at most 500 days to complete the flyover?

[Ans.: (i) 0.242, (ii) 0.8413]

23. Suppose the diameter at breast height (in.) of trees of a certain type is normally
distributed with μ = 8.8 and σ = 2.8, as suggested in the article “Simulating a Harvester-
Forwarder Softwood Thinning” (Forest Products J., May 1997: 36−41).

a. What is the probability that the diameter of a randomly selected tree will be at
least 10 in? Will exceed 10 in.? [Ans 0.336]

b. What is the probability that the diameter of a randomly selected tree will exceed 20
in.[Ans 0]

c. What is the probability that the diameter of a randomly selected tree will be
between 5 and 10 in.? [Ans 0.5795]

d. What value c is such that the interval (8.8 –c, 8.8+c) includes 98% of all diameter
values [Ans 6.524]

e. If four trees are independently selected, what is the probability that at least one has a
diameter exceeding 10 in.? [Ans 0.8028]

Binomial Approximation
1. Let X have a binomial distribution with parameters n=25 and p. calculate
each of the following probabilities using the normal approximation for the
cases p=0.5, 0.6, 0.8.

a) P(15≤ 𝑋 ≤ 20) (0.211, 0.567,

0.596)
83 | P a g e RAHUL STUDY GROUP

b) P(X≤ 15) (0.885, 0.579, 0.012)

c) P(X≥ 20) (0.003, 0.033, 0.599)

2. Suppose only 75% of all drivers in a certain state regularly wear a seat belt.
A random sample of 500 drivers is selected. What is the probability
a) b/w 360 and 400 ( inclusive) of the drivers in the sample regularly wear a
seat belt? (0.9409)
b) Fewer than 400 of those in the sample regularly wear a seat belt?
(0.9943)

3. In response to concerns about nutritional contents of fast foods,

McDonald’s has announced that it will use a new cooking oil for its
french fries that will decrease substantially trans fatty acid levels
and increase the amount of more beneficial polyunsaturated fat.
The company claims that 97 out of 100 people cannot detect a
difference in taste between the new and old oils. Assuming that
this figure is correct (as a long-run proportion), what is the
approximate probability that in a random sample of 1000
individuals who have purchased fries at McDonald’s,
a) At least 40 can taste the difference between the two oils? [Ans 0.0392
]
b) At most 5% can taste the difference between the two oils? [Ans 1.00 ]

4. Let X denote the number of flaws along a 100-m reel of magnetic

tape (an integer-valued variable). Suppose X has approximately a
normal distribution with μ = 25 and σ = 5. Use the continuity
correction to calculate the probability that the number of flaws is
a) Between 20 and 30, inclusive. [Ans 0.7286 ]
b) At most 30. Less than 30. [Ans .8643 , .8159 ]

5. Suppose that 10% of all steel shafts produced by a certain process

are nonconforming but can be reworked (rather than having to be
scrapped). Consider a random sample of 200 shafts, and let X
denote the number among these that are nonconforming and can
be reworked. What is the (approximate) probability that X is
a) At most 30? [Ans 0.9932 ]
b) Less than 30? [Ans .9875 ]
c) Between 15 and 25 (inclusive)? [Ans .8064 ]
84 | P a g e RAHUL STUDY GROUP

Miscellaneous
1. Let{𝑋𝑖 } be a sequence of i.i.d random variables such that 𝑋𝑖 = 1with probability p
and 0 with probability 1 – p
1if ∑ni=1 Xi = 100 𝒏
Define y= { } then 𝐸(𝑦 2 ) [Ans ( ) 𝒑𝟏𝟎𝟎 (𝟏 − 𝒑)𝒏−𝟏𝟎𝟎 ]
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝟏𝟎𝟎

2. Let X be a Normally distributed random variable with mean 0 and variance 1. Let
Φ(.) be the cumulative distribution function of the variable X. Then the expectation
of Φ(X) [ Ans ½]

3. Two independent variables X and Y are uniformly distributed in the interval [0, 1].
For a z ∈ [0, 1], we told that probability that max(X, Y) ≤ 𝑧 is equal to the
probability that min(X, Y) ≤ (1 − 𝑧). What is the value of z? [Ans 1/√𝟐 ]

4. A Random variable X has the Standard normal distribution . what is the probability
that X lies in the interval [2 ,3 ]
5. If a binomial random variable X has expectation 7 and variance 2.1, then the
probability that X = 11 [Ans 0]
6. The vitamin content of a particular brand of vitamin supplement pills is normally
distributed with mean 490 mg and standard deviation 12 mg. What is the
probability (approximately) that a randomly selected pill contains at least 500 mg of
Vitamin C? [Ans 0.2]

7. Suppose X has a normal distribution with mean 0 and variance σ 2 . Let Y be an

independent random variable taking values -1 and 1 with equal probability. Define
Z=XY + X/Y. Show that Var(Z)>4 σ2

8. What is the probability that atleast one six appears when six fair dice are rolled ?
5 6
[Ans 1- (6) ]

9. A random variable X ,Uniformly distributed on [0,1] ,divides [0,1] into two segments
of length X and (1-X) . Let R be the ratio of the smaller to the larger segments ( i.e R
85 | P a g e RAHUL STUDY GROUP

= X/(1-X) , or R = (1-X)/X , Depending on whether X ≤ ½ or X >1/2. The

distribution of R , F (r ) that is the probability that R ≤r . [Ans 2r/ (r+1) ]
∞ 2 /2
10. Solve using Normal Distribution ∫1.96 𝑒 −𝑥 𝑑𝑥 [Ans .025√𝟐𝞟 ]

11. Out of 800 families with five children each, how many families would you expect to
have either 2 or 3 boys? Assume equal probabilities for boys and girls [Ans 500 ]

12. Suppose that 30% of all students who have to buy a text for a particular course want
a new copy (the successes!), whereas the other 70% want a used copy. Consider
randomly selecting 25 purchasers..

a) What are the mean value and standard deviation of the number who want a
new copy of the book?
b) What is the probability that the number who want new copies is more than two
standard deviations away from the mean value?
c) The bookstore has 15 new copies and 15 used copies in stock. If 25 people come
in one by one to purchase this text, what is the probability that all 25 will get the
type of book they want from current stock?
d) . Suppose that new copies cost $100 and used copies cost $70. Assume the
bookstore currently has 50 new copies and 50 used copies. What is the expected
value of total revenue from the sale of the next 25 copies purchased? [Ans 1975
]

13. Customers at a gas station pay with a credit card, a debit card, or cash. Assume
that successive customers make independent choices, with P( credit card ) = .5,
P( debit card ) = .2, P( cash ) = .3. Among the next 100 customers, what are the
mean and variance of the number who pay with a credit card? [Ans 50 ,25 ]

14. A friend recently planned a camping trip. He had two flashlights, one that
required a single 6-V battery and another that used two size-D batteries. He had
previously packed two 6-V and four size-D batteries in his camper. Suppose that
theprobability than any particular battery works is p and that batteries work or
fail independently of one another. Our friend wants to take just one flashlight.
For what values of p should he take the6-V flashlight?

1.
15. A k out of n system is one in which there is a group of n components, and
the system will function if at least k of the components function. Assume the
components function independently of one another. ... In a 3 out of 5 system,
86 | P a g e RAHUL STUDY GROUP

each component has probability 0.9 ,What is the probability that a 3 out of 5
system of functions ?[Ans.9913 ]

16. Of all customers purchasing automatic garage-door openers, 75% purchase a

chain-driven model. Let X = the number among the next 15 purchasers who
select the chain-driven model.

a. What is the pmf of X? [Ans Binomial ]

b. Compute P(X > 10). [Ans .852 ]

c. Compute P(6 ≤ X ≤ 10). [Ans

d. Compute μ and 𝜎 2 .

e. If the store currently has in stock 10 chain-driven models and 8 shaft-driven

models, what is the probability that the requests of these 15 customers can all be
met from existing stock?

17. . Forty percent of seeds from maize (modern-day corn) ears carry single
spikelets, and the other 60% carry paired spikelets. A seed with single spikelets
will produce an ear with single spikelets 29 % of the time, whereas a seed with
paired spikelets will produce an ear with single spikelets 26% of the time.
Consider randomly selecting ten seeds.
(a) What is the probability that exactly five of these seeds carry a single spikelet
and produce an ear with a single spikelet? [Ans .002857 ]
b) What is the probability that exactly five of the ears produced by these seeds
have single spikelets? What is the probability that at most five ears have single
spikelets? [Ans .97024 ]

18. A trial has just resulted in a hung jury because 8 members of the jury were in
favour of a guilty verdict and the other 4 for acquittal. If the jurors leave the jury
room in random order and each of the first four leaving the room is accosted by a
reporter in quest of an interview,
a. what is the probability mass function of X, the number of jurors favoring
acquittal among those interviewed? [Ans Hypergeometric ]
b. How many of those favouring acquittal do you expect to be interviewed?

19. A reservation service employs five information operators. Those information

operators receive requests for information independently of one another,each
according to a Poisson process with rate λ = 2 per minute.
a) What is the probability that during a given 1-min period,the first operator
receives no requests ?
87 | P a g e RAHUL STUDY GROUP

b) What is the probability that during a given 1-min period, exactly four of the
five operators receive no requests ?

20. Let X=the time it takes a read/write head to locate a desired record on a
computer disk memory device once the head has been positioned over the
correct track. If the disks rotate once every 25 millisec, a reasonable assumption
is that X is uniformly distributed on the interval [0, 25].
a) Compute P(10 ≤ X ≤ 20) [Ans 0.4 ]
b) Compute P(X ≥ 20) [Ans 0.6 ]
c) Obtain the cdf F(x).
d) Compute E(x) and σx [Ans 12.5 , 52.083 ]
e) Find the Median

21. The breakdown voltage of a randomly chosen diode of a certain type is known to
be normally distributed with mean value 40 V and standard deviation 1.5 V.
a) What is the probability that the voltage of a single diode is between 39 and 42?
[Ans 0.6568 ]
b) What value is such that only 15% of all diodes have voltages exceeding that
value? [Ans 41.56 ]
c) If four diodes are independently selected, what is the probability that at least
one has a voltage exceeding 42? [Ans .3197 ]

22. The article “Computer Assisted Net Weight Control”(Quality Progress, 1983: 22–
25) suggests a normal distribution with mean 137.2 oz and standard deviation
1.6 oz for the actual contents of jars of a certain type. The stated contents was
135 oz.
a. What is the probability that a single jar contains more than the stated
contents? [Ans .9162 ]
b. Among ten randomly selected jars, what is the probability that at least
eight contain more than the stated contents? [Ans .9549 ]
c. Assuming that the mean remains at 137.2, to what value would the
standard deviation have to be changed so that 95% of all jars contain
more than the stated contents? [Ans 1.33 ]

23. When circuit boards used in the manufacture of compact disc players are tested,
the long-run percentage of defectives is 5%. Suppose that a batch of 250 boards
has been received and that the condition of any particular board is independent
of that of any other board.
a) What is the approximate probability that at least 10% of the boards in the batch
are defective? [Ans .0003 ]
b) What is the approximate probability that there are exactly 10 defectives in the
batch? [ Ans .0888]

24. Suppose that X has a Normal distribution such that P(X < 116) =0.20 and
P(X<328) =0.90. Determine the mean and variance of X.
88 | P a g e RAHUL STUDY GROUP

[Ans 200 , 10000]

25. The Lifetime X of an electronic component has the exponential distribution such
that P(X ≤1000) =0.75. what is the expected lifetime of the component ? [Ans
1000/Log(0.25) ]

26. Two companies A & B drill wells in a rural area. Company A charges a flat fee of
Rs. 3500 to drill a well regardless of its depth. Company B charges Rs. 1000 plus
Rs. 12 per ft to drill a well. The depths of wells drilled in this area have a normal
distribution with a mean of 250 ft. and a standard deviation of 40 ft.
(i) What is the probability that company B would charge more than company
A to drill a well?[ Ans:.8508]

(ii) Find the mean amount charged by company B to drill a well.[ Ans: 4000]

27. Suppose that student participation in a competition that happens every year has a
normal distribution with mean 104 students and standard deviation 5 students.
What is the probability that student participation differs from mean by more than
1 standard deviation .How would you characterize the top extreme 0.1% of the
student participation values?[Ans 0.1587 ,119.4]

Joint probability distribution

DISCERETE PROB DIST

1)
X Y→ 0 1 2 3 4
↓
0 0.08 0.07 0.06 0.01 0.01
89 | P a g e RAHUL STUDY GROUP

1 0.06 0.10 0.12 0.05 0.02

2 0.05 0.06 0.09 0.04 0.03

3 0.02 0.03 0.03 0.03 0.04

Determine each of the following probabilities:

a) P(X=2 ) (0.25)

b) P(Y ≥ 2) ( 0.53)

c) P(X ≤ 2 & 𝑌 ≤ 2) (0.69)

d) P(X=Y) (0.3)

e) P(X > 𝑌) (0.25)

2) A service station has both self – service and full – service islands.
On each island, there is a single regular unleaded pump with two
hoses. Let X denote the number of hoses being used on the self –
service island at a particular time, and let Y denote the number of
hoses on the full – service island in use at that time. The joint pmf
of X and Y appears in the accompanying tabulation.

p(x↓, y→) 0 1 2

0 0.10 0.04 0.02

1 0.08 0.20 0.06

2 0.06 0.14 0.30

90 | P a g e RAHUL STUDY GROUP

a) What is P(X = 1 and Y= 1)? (0.20)

b) Compute P(X ≤ 1 and Y ≤ 1)? (0.42)
c) Give a word description of the event {𝑋 ≠ 0 𝑎𝑛𝑑 𝑌 ≠ 0 } and compute
the probability of this event. (0.70)
d) Compute the marginal pmf of X and Y?

X P(x)

0 0.16

1 0.34

2 0.50

Y P(y)

0 0.24

1 0.38

2 0.38

e) Are X and Y independent rv’s? Explain. (No)

3) When a automobile is stopped by roving safety patrol, each tire is

checked for tire wire, and each headlight is checked to see
whether it is properly aimed. Let X denotes the number of
headlights that need adjustment, and let Y denote the number of
defective tires.

a) If X and Y independent with p𝑥 (0) = 0.5, p𝑥 (1) = 0.3, p𝑥 (2) = 0.2,

and p𝑦 (0) = 0.6, p𝑦 (1) = 0.1, p𝑦 (2) = p𝑦 (3) = 0.05 and p𝑦 (4) =
0.2, display the joint pmf of (X, Y) in a joint probability table.

P(x↓, y→) 0 1 2 3 4
91 | P a g e RAHUL STUDY GROUP

0 0.30 0.05 0.025 0.025 0.10 0.5

1 0.18 0.03 0.015 0.015 0.06 0.3

2 0.12 0.02 0.01 0.01 0.04 0.2

0.6 0.1 0.05 0.05 0.2

b) Compute P(X ≤ 1 and Y ≤ 1) from the joint probability table and

verify that it equals the product P(X ≤ 1). P(Y ≤ 1). (0.56)

c) What is P(X + Y = 0)(the probability of no violations)? (0.30)

d) Compute P(X + Y ≤ 1). (0.53)

4) A certain market has both an express checkout line and a super

express checkout line. Let X denote the number of customers in
line at the express checkout at a particular time of day, and let Y
denote the number of customers in line at the super express
checkout at the same time. Suppose the joint pmf of X and Y is as
given in the accompanying table.

(X↓, Y→) 0 1 2 3

0 0.08 0.07 0.04 0.00

1 0.06 0.15 0.05 0.04

2 0.05 0.04 0.10 0.06

3 0.00 0.03 0.04 0.07

4 0.00 0.01 0.05 0.06

a) What is P(X = 1 and Y= 1), that is, the probability that there is
exactly one customer in each line? (0.15)
92 | P a g e RAHUL STUDY GROUP

b) What is P(X = Y), that is, the probability that the numbers of
customers in the two lines are identical? (0.40)
c) Let A denote the event that there are at least two more customers in
one line than in the other line. Express A in terms of X & Y, and
calculate the probability of this event. (0.22)
d) What is the probability that the total number of customers in the two
lines is exactly four? At least four? (0.17); (0.46)

5)

X Y→ 1 2 3
↓
1 0.10 0.10 0.05

2 0.15 0.10 0.05

3 0.20 0.05 -

4 0.15 0.05 -

a) Determine P(X=1), P(X=2)and P(X=3)?

P(X=1)=0.6
P(X=2)=0.3
P(X=3)=0.1
b) Find the marginal probability distribution of X?

X 1 2 3

P(X=x) 0.6 0.3 0.1

c) Find the marginal probability distribution of Y?

Y 1 2 3 4
93 | P a g e RAHUL STUDY GROUP

P(Y=y) 0.25 0.3 0.25 0.2

6) Obtain the probability function for the marginal distribution of N from

0, where

𝑚
P(M=m, N=n)= where m=1, 2, 3, 4 and n=1, 2, 3?
35×2𝑛−2

a) P(M=3, N= 1 or 2) (9/35)

b) P(N=3) (1/7)

c) P(M=2/N=3) (1/5)

d) Also, check whether M & N are independent or not?

7)

X Y→ 10 20 30
↓
1 0.2 0.2 0.1

2 0.2 0.3 0

a) Calculate E(Y/X=1)? (18)

b) Calculate E(X/Y=1)? (1.5)

c) Calculate variance(Y/X=1)? (56)

8) Suppose that X & Y have a discrete distribution for which the joint
PDF is as follow:
94 | P a g e RAHUL STUDY GROUP

𝑐|𝑋 + 𝑌| , 𝑓𝑜𝑟𝑥 = −2, −1, 0, 1, 2

f(X, Y)={ 𝑓𝑜𝑟 𝑦 = −2, −1, 0,1,2
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

a) Determine the value of c? (1/40)

b) P(X=0 & Y= -2) (1/20)

c) P(X=1) (7/40)

d) P(|𝑋 − 𝑌| ≤ 1) (0.7)

9)
Y X→ 0 1 2
↓
1 0.1 0.1 0

2 0.1 0.1 0.2

3 0.2 0.1 0.1

a) The marginal distribution of X?

X 0 1 2

P(X=x) 0.4 0.3 0.3

b) The conditional distribution of X/Y=2?

P(X=0/Y=2)=0.25
P(X=1/Y=2)=0.25
P(X=2/Y=2)=0.5

c) Are X and Y are independent?

95 | P a g e RAHUL STUDY GROUP

d) Calculate the expected value of 𝑋 2 + 2𝑌? (5.5)

e) Also, verify that E[𝑋 2 + 2𝑋𝑌] = 𝐸[𝑋 2 ] + 𝐸[2𝑌]?

f) Calculate the covariance of the random variables X & Y? (0.02)

10) Suppose that X & Y have a joint discrete distribution for which the joint
PDF is defined as follow:

1
(𝑋 + 𝑌), 𝑓𝑜𝑟 𝑋 = 0, 1, 2
30
f(X, Y)= { 𝑓𝑜𝑟 𝑌 = 0, 1, 2, 3
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

a) Determine the marginal distribution of X & Y?

b) X & Y independent?

11)

N M→ 1 2 3 4
↓
1 2/35 4/35 6/35 8/35

2 1/35 2/35 3/35 4/35

3 1/70 1/35 3/70 2/35

𝑁+1
a) Calculate the expected value of ? (36/35)
𝑀

𝑁+1 1
b) Also, verify that E[ ] = 𝐸 [ ] 𝐸[𝑁 + 1]
𝑀 𝑀
96 | P a g e RAHUL STUDY GROUP

12) Each cell of the following table provides the probability of the joint
occurrence of the corresponding pair of values of the random
variables X and Y.

X↓ Y→ 1 2 3 4

1 .1 0 .1 0

2 .3 0 .1 .2

3 0 .1 0 .1

Consider the following statements about X and Y:

I. Pr(Y = 2) >Pr(X = 1)
II. Pr(Y = 1|X = 2) = Pr(Y = 1|X =1)
III. The events X = 3 and Y = 3 are mutually exclusive.
IV. X and Y are independent.

Which of the above statements are true?

a) Only I and II.

b) Only II and III.
c) Only III and IV.
d) Only II, III and IV.
13) Suppose X and Y are two discrete random variables which have the joint
probability mass function p(x,y)=(x+2y)/18,(x,y)=(1,1),(1,2),(2,1),(2,2), 0
elsewhere. Determine the conditional mean of Y, given X=2. Also find the two
marginal Probability mass function. Calculate the value of E(3X-2Y

14) The joint probability distribution of the number X of cars and the
number Y of buses per signal cycle at a proposed left-turn lane is
displayed in the accompanying joint probability table.
97 | P a g e RAHUL STUDY GROUP

a. What is the probability that there is exactly one car and exactly
one bus during a cycle? [Ans 0.030 ]
b. What is the probability that there is at most one car and at
most one bus during a cycle? [Ans .120 ]
c. What is the probability that there is exactly one car during a
cycle? Exactly one bus? [Ans .300 ]
d. Suppose the left-turn lane is to have a capacity of five cars, and
that one bus is equivalent to three cars. What is the probability
of an overflow during a cycle? [Ans 0.380 ]
e. Are X and Y independent rv’s? Explain . [Ans Yes ]
15) An instructor has given a short quiz consisting of two parts. For a randomly selected
student, let X = the number of points earned on the first part and Y = the number of
points earned on the second part. Suppose that the joint pmf of X and Y is given in
the accompanying table.

Y
p(x,y) 0 5 10 15
0 0.02 0.06 0.02 0.10
X 5 0.04 0.14 0.20 0.10
10 0.01 0.15 0.15 0.01

a. If the score recorded in the grade book is the total number of points earned on the two
parts, what is the expected recorded score E(X + Y)? [Ans 14.10 ]
b. If the maximum of the two scores is recorded, what is the expected recorded score? [Ans
9.10 ]

Continuous Prob Distribution

1. The continuous random variables U and V have the joint probability density function
1
f(u ,v) = (2𝑢 + 𝑣) where 10 <u <20 , -5 < v < 5
3000

a. Calculate P(10 < U <15 , V>0 )

b. Calculate Marginal Densities function of U and V. [Ans 0.229 ]
c. Calculate E(U ) , E(V) , V(U) , V(V) ? [140/9 ,5/18 , 650/81 , 2675/ 324]
d. Calculate correlation Cofficient between U and V [Ans -.019 ]
98 | P a g e RAHUL STUDY GROUP

2. The continuous random variables U and V have the joint probability density function
1
f(X ,Y) = (𝑥 + 3𝑦) where 0 <X <2 , 0 < y < 2
16
a. Determine the conditional density function of X given Y=y
b. Determine the Conditional Expectation function of X given Y=y
c. Calculate Expectation and Variance of X and Y ?
d. Calculate Correlation Cofficient between X and Y ?

3. Let X and Y have Joint density function given by :

f(x ,y) = C(x+ 3y) 0<x<2 ,0 <y < 2

a. Calculate the value of c. [Ans 1/16 ]

b. Hence calculate P( X < 1 , Y >0.5 ) [Ans 0.398 ]

4. The continuous random variables X and Y have the joint probability density function
4
f(X ,Y) = (3𝑥 2+xy) where 0 <X <1 , 0 < y < 1
5

a. Find marginal Density Function of X

b. The Conditional density function of Y given X=x
c. The covariance between X and Y [Ans 11/15 ]

5. The continuous random variables X and Y have the joint probability density function
f(X ,Y) = 24xy where 0 <X <1 , 0 < y < 1 , 0 < x+y < 1

a. Check whether function is well defined or not ?

b. Find the P( x+y < 0.5 ) [Ans 0.0625]
c. Find the Marginal distribution
d. Find the Covariance . [Ans -2/75 ]

6. Each front tire on a particular type of vehicle is supposed to be

filled to a pressure of 26 psi. Suppose the actual air pressure in
each tire is a random variable X for the right tire and Y for the left
tire, with joint pdf
99 | P a g e RAHUL STUDY GROUP

a) What is the value of K? [Ans 3/3800000]

b) What is the probability that both tires are underfilled? [Ans .3024 ]
c) What is the probability that the difference in air pressure
between the two tires is at most 2 psi? [Ans .3593 ]
d) Determine the (marginal) distribution of air pressure in the right tire
alone. [Ans f(x) = 10k𝑥 2 + .05 , f(y) = 10k𝑦 2 + .05 ]
e) Are X and Y independent rv’s? [Ans No]

7. Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M.
for dinner at a local health-food restaurant. Let X=Annie’s arrival time
and
Y=Alvie’s arrival time. Suppose X and Y are independent with each
uniformly distributed on the interval [5, 6].
a) What is the joint pdf of X and Y?
b) What is the probability that they both arrive between 5:15 and 5:45?
[Ans .25]
c) If the first one to arrive will wait only 10 min before leaving to eat
elsewhere, what is the probability that they have dinner at the
health-food restaurant? [Ans .306 ]
 A-2 Appendix Tables

x
Table A.1 Cumulative Binomial Probabilities B(x; n, p) 5 g b(y; n, p)
a. n ⴝ 5 y50

0.01 0.05 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0.99

0 .951 .774 .590 .328 .237 .168 .078 .031 .010 .002 .001 .000 .000 .000 .000
1 .999 .977 .919 .737 .633 .528 .337 .188 .087 .031 .016 .007 .000 .000 .000
x 2 1.000 .999 .991 .942 .896 .837 .683 .500 .317 .163 .104 .058 .009 .001 .000
3 1.000 1.000 1.000 .993 .984 .969 .913 .812 .663 .472 .367 .263 .081 .023 .001
4 1.000 1.000 1.000 1.000 .999 .998 .990 .969 .922 .832 .763 .672 .410 .226 .049

b. n ⴝ 10

0.01 0.05 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0.99

0 .904 .599 .349 .107 .056 .028 .006 .001 .000 .000 .000 .000 .000 .000 .000
1 .996 .914 .736 .376 .244 .149 .046 .011 .002 .000 .000 .000 .000 .000 .000
2 1.000 .988 .930 .678 .526 .383 .167 .055 .012 .002 .000 .000 .000 .000 .000
3 1.000 .999 .987 .879 .776 .650 .382 .172 .055 .011 .004 .001 .000 .000 .000
4 1.000 1.000 .998 .967 .922 .850 .633 .377 .166 .047 .020 .006 .000 .000 .000
x
5 1.000 1.000 1.000 .994 .980 .953 .834 .623 .367 .150 .078 .033 .002 .000 .000
6 1.000 1.000 1.000 .999 .996 .989 .945 .828 .618 .350 .224 .121 .013 .001 .000
7 1.000 1.000 1.000 1.000 1.000 .998 .988 .945 .833 .617 .474 .322 .070 .012 .000
8 1.000 1.000 1.000 1.000 1.000 1.000 .998 .989 .954 .851 .756 .624 .264 .086 .004
9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .994 .972 .944 .893 .651 .401 .096

c. n ⴝ 15

0.01 0.05 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0.99

0 .860 .463 .206 .035 .013 .005 .000 .000 .000 .000 .000 .000 .000 .000 .000
1 .990 .829 .549 .167 .080 .035 .005 .000 .000 .000 .000 .000 .000 .000 .000
2 1.000 .964 .816 .398 .236 .127 .027 .004 .000 .000 .000 .000 .000 .000 .000
3 1.000 .995 .944 .648 .461 .297 .091 .018 .002 .000 .000 .000 .000 .000 .000
4 1.000 .999 .987 .836 .686 .515 .217 .059 .009 .001 .000 .000 .000 .000 .000
5 1.000 1.000 .998 .939 .852 .722 .403 .151 .034 .004 .001 .000 .000 .000 .000
6 1.000 1.000 1.000 .982 .943 .869 .610 .304 .095 .015 .004 .001 .000 .000 .000
x 7 1.000 1.000 1.000 .996 .983 .950 .787 .500 .213 .050 .017 .004 .000 .000 .000
8 1.000 1.000 1.000 .999 .996 .985 .905 .696 .390 .131 .057 .018 .000 .000 .000
9 1.000 1.000 1.000 1.000 .999 .996 .966 .849 .597 .278 .148 .061 .002 .000 .000
10 1.000 1.000 1.000 1.000 1.000 .999 .991 .941 .783 .485 .314 .164 .013 .001 .000
11 1.000 1.000 1.000 1.000 1.000 1.000 .998 .982 .909 .703 .539 .352 .056 .005 .000
12 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .973 .873 .764 .602 .184 .036 .000
13 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .965 .920 .833 .451 .171 .010
14 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .987 .965 .794 .537 .140

(continued )

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-3

x
Table A.1 Cumulative Binomial Probabilities (cont.) B(x; n, p) 5 g b(y; n, p)
y50
d. n ⴝ 20

0.01 0.05 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0.99

0 .818 .358 .122 .012 .003 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000
1 .983 .736 .392 .069 .024 .008 .001 .000 .000 .000 .000 .000 .000 .000 .000
2 .999 .925 .677 .206 .091 .035 .004 .000 .000 .000 .000 .000 .000 .000 .000
3 1.000 .984 .867 .411 .225 .107 .016 .001 .000 .000 .000 .000 .000 .000 .000
4 1.000 .997 .957 .630 .415 .238 .051 .006 .000 .000 .000 .000 .000 .000 .000
5 1.000 1.000 .989 .804 .617 .416 .126 .021 .002 .000 .000 .000 .000 .000 .000
6 1.000 1.000 .998 .913 .786 .608 .250 .058 .006 .000 .000 .000 .000 .000 .000
7 1.000 1.000 1.000 .968 .898 .772 .416 .132 .021 .001 .000 .000 .000 .000 .000
8 1.000 1.000 1.000 .990 .959 .887 .596 .252 .057 .005 .001 .000 .000 .000 .000
9 1.000 1.000 1.000 .997 .986 .952 .755 .412 .128 .017 .004 .001 .000 .000 .000
x
10 1.000 1.000 1.000 .999 .996 .983 .872 .588 .245 .048 .014 .003 .000 .000 .000
11 1.000 1.000 1.000 1.000 .999 .995 .943 .748 .404 .113 .041 .010 .000 .000 .000
12 1.000 1.000 1.000 1.000 1.000 .999 .979 .868 .584 .228 .102 .032 .000 .000 .000
13 1.000 1.000 1.000 1.000 1.000 1.000 .994 .942 .750 .392 .214 .087 .002 .000 .000
14 1.000 1.000 1.000 1.000 1.000 1.000 .998 .979 .874 .584 .383 .196 .011 .000 .000
15 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .994 .949 .762 .585 .370 .043 .003 .000
16 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .984 .893 .775 .589 .133 .016 .000
17 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .965 .909 .794 .323 .075 .001
18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .992 .976 .931 .608 .264 .017
19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .997 .988 .878 .642 .182

(continued)

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-4 Appendix Tables

x
Table A.1 Cumulative Binomial Probabilities (cont.) B(x; n, p) 5 g b(y; n, p)
y50
e. n ⴝ 25

0.01 0.05 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 0.95 0.99

0 .778 .277 .072 .004 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
1 .974 .642 .271 .027 .007 .002 .000 .000 .000 .000 .000 .000 .000 .000 .000
2 .998 .873 .537 .098 .032 .009 .000 .000 .000 .000 .000 .000 .000 .000 .000
3 1.000 .966 .764 .234 .096 .033 .002 .000 .000 .000 .000 .000 .000 .000 .000
4 1.000 .993 .902 .421 .214 .090 .009 .000 .000 .000 .000 .000 .000 .000 .000
5 1.000 .999 .967 .617 .378 .193 .029 .002 .000 .000 .000 .000 .000 .000 .000
6 1.000 1.000 .991 .780 .561 .341 .074 .007 .000 .000 .000 .000 .000 .000 .000
7 1.000 1.000 .998 .891 .727 .512 .154 .022 .001 .000 .000 .000 .000 .000 .000
8 1.000 1.000 1.000 .953 .851 .677 .274 .054 .004 .000 .000 .000 .000 .000 .000
9 1.000 1.000 1.000 .983 .929 .811 .425 .115 .013 .000 .000 .000 .000 .000 .000
10 1.000 1.000 1.000 .994 .970 .902 .586 .212 .034 .002 .000 .000 .000 .000 .000
11 1.000 1.000 1.000 .998 .980 .956 .732 .345 .078 .006 .001 .000 .000 .000 .000
x 12 1.000 1.000 1.000 1.000 .997 .983 .846 .500 .154 .017 .003 .000 .000 .000 .000
13 1.000 1.000 1.000 1.000 .999 .994 .922 .655 .268 .044 .020 .002 .000 .000 .000
14 1.000 1.000 1.000 1.000 1.000 .998 .966 .788 .414 .098 .030 .006 .000 .000 .000
15 1.000 1.000 1.000 1.000 1.000 1.000 .987 .885 .575 .189 .071 .017 .000 .000 .000
16 1.000 1.000 1.000 1.000 1.000 1.000 .996 .946 .726 .323 .149 .047 .000 .000 .000
17 1.000 1.000 1.000 1.000 1.000 1.000 .999 .978 .846 .488 .273 .109 .002 .000 .000
18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .993 .926 .659 .439 .220 .009 .000 .000
19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .971 .807 .622 .383 .033 .001 .000
20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .910 .786 .579 .098 .007 .000
21 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .967 .904 .766 .236 .034 .000
22 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .968 .902 .463 .127 .002
23 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .993 .973 .729 .358 .026
24 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .996 .928 .723 .222

x
e2m my
Table A.2 Cumulative Poisson Probabilities F(x; m) 5 g y!
y50

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

0 .905 .819 .741 .670 .607 .549 .497 .449 .407 .368
1 .995 .982 .963 .938 .910 .878 .844 .809 .772 .736
2 1.000 .999 .996 .992 .986 .977 .966 .953 .937 .920
x 3 1.000 1.000 .999 .998 .997 .994 .991 .987 .981
4 1.000 1.000 1.000 .999 .999 .998 .996
5 1.000 1.000 1.000 .999
6 1.000

(continued)

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-5

x
e2mmy
Table A.2 Cumulative Poisson Probabilities (cont.) F(x; m) 5 g
y50 y!

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 15.0 20.0

0 .135 .050 .018 .007 .002 .001 .000 .000 .000 .000 .000
1 .406 .199 .092 .040 .017 .007 .003 .001 .000 .000 .000
2 .677 .423 .238 .125 .062 .030 .014 .006 .003 .000 .000
3 .857 .647 .433 .265 .151 .082 .042 .021 .010 .000 .000
4 .947 .815 .629 .440 .285 .173 .100 .055 .029 .001 .000
5 .983 .916 .785 .616 .446 .301 .191 .116 .067 .003 .000
6 .995 .966 .889 .762 .606 .450 .313 .207 .130 .008 .000
7 .999 .988 .949 .867 .744 .599 .453 .324 .220 .018 .001
8 1.000 .996 .979 .932 .847 .729 .593 .456 .333 .037 .002
9 .999 .992 .968 .916 .830 .717 .587 .458 .070 .005
10 1.000 .997 .986 .957 .901 .816 .706 .583 .118 .011
11 .999 .995 .980 .947 .888 .803 .697 .185 .021
12 1.000 .998 .991 .973 .936 .876 .792 .268 .039
13 .999 .996 .987 .966 .926 .864 .363 .066
14 1.000 .999 .994 .983 .959 .917 .466 .105
15 .999 .998 .992 .978 .951 .568 .157
16 1.000 .999 .996 .989 .973 .664 .221
17 1.000 .998 .995 .986 .749 .297
18 .999 .998 .993 .819 .381
x
19 1.000 .999 .997 .875 .470
20 1.000 .998 .917 .559
21 .999 .947 .644
22 1.000 .967 .721
23 .981 .787
24 .989 .843
25 .994 .888
26 .997 .922
27 .998 .948
28 .999 .966
29 1.000 .978
30 .987
31 .992
32 .995
33 .997
34 .999
35 .999
36 1.000

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-6 Appendix Tables

Table A.3 Standard Normal Curve Areas Φ(z)  P(Z  z)

Standard normal density curve

Shaded area = Φ(z)

0 z

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

⫺3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002
⫺3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003
⫺3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005
⫺3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007
⫺3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010
⫺2.9 .0019 .0018 .0017 .0017 .0016 .0016 .0015 .0015 .0014 .0014
⫺2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019
⫺2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026
⫺2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036
⫺2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0038
⫺2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064
⫺2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084
⫺2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110
⫺2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143
⫺2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183
⫺1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233
⫺1.8 .0359 .0352 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294
⫺1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367
⫺1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455
⫺1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .0559
⫺1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0722 .0708 .0694 .0681
⫺1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823
⫺1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985
⫺1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170
⫺1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379
⫺0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611
⫺0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867
⫺0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148
⫺0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451
⫺0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776
⫺0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121
⫺0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3482
⫺0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859
⫺0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247
⫺0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641
(continued)

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-7

Table A.3 Standard Normal Curve Areas (cont.) ⌽(z) 5 P(Z # z)

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9278 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-8 Appendix Tables

x
Table A.4 The Incomplete Gamma Function 1 a21 2y
F(x; a) 5 冮 y e dy
0
⌫(a)
x␣ 1 2 3 4 5 6 7 8 9 10

1 .632 .264 .080 .019 .004 .001 .000 .000 .000 .000
2 .865 .594 .323 .143 .053 .017 .005 .001 .000 .000
3 .950 .801 .577 .353 .185 .084 .034 .012 .004 .001
4 .982 .908 .762 .567 .371 .215 .111 .051 .021 .008
5 .993 .960 .875 .735 .560 .384 .238 .133 .068 .032
6 .998 .983 .938 .849 .715 .554 .394 .256 .153 .084
7 .999 .993 .970 .918 .827 .699 .550 .401 .271 .170
8 1.000 .997 .986 .958 .900 .809 .687 .547 .407 .283
9 .999 .994 .979 .945 .884 .793 .676 .544 .413
10 1.000 .997 .990 .971 .933 .870 .780 .667 .542
11 .999 .995 .985 .962 .921 .857 .768 .659
12 1.000 .998 .992 .980 .954 .911 .845 .758
13 .999 .996 .989 .974 .946 .900 .834
14 1.000 .998 .994 .986 .968 .938 .891
15 .999 .997 .992 .982 .963 .930

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-9

Table A.5 Critical Values for t Distributions t density curve

Shaded area = 

0 t,

v .10 .05 .025 .01 .005 .001 .0005

1 3.078 6.314 12.706 31.821 63.657 318.31 636.62

2 1.886 2.920 4.303 6.965 9.925 22.326 31.598
3 1.638 2.353 3.182 4.541 5.841 10.213 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.893 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 1.319 1.714 2.069 2.500 2.807 3.485 3.767
24 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.450 3.725
26 1.315 1.706 2.056 2.479 2.779 3.435 3.707
27 1.314 1.703 2.052 2.473 2.771 3.421 3.690
28 1.313 1.701 2.048 2.467 2.763 3.408 3.674
29 1.311 1.699 2.045 2.462 2.756 3.396 3.659
30 1.310 1.697 2.042 2.457 2.750 3.385 3.646
32 1.309 1.694 2.037 2.449 2.738 3.365 3.622
34 1.307 1.691 2.032 2.441 2.728 3.348 3.601
36 1.306 1.688 2.028 2.434 2.719 3.333 3.582
38 1.304 1.686 2.024 2.429 2.712 3.319 3.566
40 1.303 1.684 2.021 2.423 2.704 3.307 3.551
50 1.299 1.676 2.009 2.403 2.678 3.262 3.496
60 1.296 1.671 2.000 2.390 2.660 3.232 3.460
120 1.289 1.658 1.980 2.358 2.617 3.160 3.373
∞ 1.282 1.645 1.960 2.326 2.576 3.090 3.291

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 A-10
Table A.6 Tolerance Critical Values for Normal Population Distributions

Two-sided Intervals One-sided Intervals

Confidence Level 95% 99% 95% 99%
% of Population Captured ⱖ 90% ⱖ 95% ⱖ 99% ⱖ 90% ⱖ 95% ⱖ 99% ⱖ 90% ⱖ 95% ⱖ 99% ⱖ 90% ⱖ 95% ⱖ 99%
Appendix Tables

2 32.019 37.674 48.430 160.193 188.491 242.300 20.581 26.260 37.094 103.029 131.426 185.617
3 8.380 9.916 12.861 18.930 22.401 29.055 6.156 7.656 10.553 13.995 17.370 23.896
4 5.369 6.370 8.299 9.398 11.150 14.527 4.162 5.144 7.042 7.380 9.083 12.387
5 4.275 5.079 6.634 6.612 7.855 10.260 3.407 4.203 5.741 5.362 6.578 8.939
6 3.712 4.414 5.775 5.337 6.345 8.301 3.006 3.708 5.062 4.411 5.406 7.335
7 3.369 4.007 5.248 4.613 5.488 7.187 2.756 3.400 4.642 3.859 4.728 6.412
8 3.136 3.732 4.891 4.147 4.936 6.468 2.582 3.187 4.354 3.497 4.285 5.812
9 2.967 3.532 4.631 3.822 4.550 5.966 2.454 3.031 4.143 3.241 3.972 5.389
10 2.839 3.379 4.433 3.582 4.265 5.594 2.355 2.911 3.981 3.048 3.738 5.074
11 2.737 3.259 4.277 3.397 4.045 5.308 2.275 2.815 3.852 2.898 3.556 4.829
12 2.655 3.162 4.150 3.250 3.870 5.079 2.210 2.736 3.747 2.777 3.410 4.633
13 2.587 3.081 4.044 3.130 3.727 4.893 2.155 2.671 3.659 2.677 3.290 4.472
14 2.529 3.012 3.955 3.029 3.608 4.737 2.109 2.615 3.585 2.593 3.189 4.337
15 2.480 2.954 3.878 2.945 3.507 4.605 2.068 2.566 3.520 2.522 3.102 4.222
16 2.437 2.903 3.812 2.872 3.421 4.492 2.033 2.524 3.464 2.460 3.028 4.123
Sample Size n 17 2.400 2.858 3.754 2.808 3.345 4.393 2.002 2.486 3.414 2.405 2.963 4.037
18 2.366 2.819 3.702 2.753 3.279 4.307 1.974 2.453 3.370 2.357 2.905 3.960
19 2.337 2.784 3.656 2.703 3.221 4.230 1.949 2.423 3.331 2.314 2.854 3.892
20 2.310 2.752 3.615 2.659 3.168 4.161 1.926 2.396 3.295 2.276 2.808 3.832
25 2.208 2.631 3.457 2.494 2.972 3.904 1.838 2.292 3.158 2.129 2.633 3.601
30 2.140 2.549 3.350 2.385 2.841 3.733 1.777 2.220 3.064 2.030 2.516 3.447
35 2.090 2.490 3.272 2.306 2.748 3.611 1.732 2.167 2.995 1.957 2.430 3.334
40 2.052 2.445 3.213 2.247 2.677 3.518 1.697 2.126 2.941 1.902 2.364 3.249
45 2.021 2.408 3.165 2.200 2.621 3.444 1.669 2.092 2.898 1.857 2.312 3.180
50 1.996 2.379 3.126 2.162 2.576 3.385 1.646 2.065 2.863 1.821 2.269 3.125
60 1.958 2.333 3.066 2.103 2.506 3.293 1.609 2.022 2.807 1.764 2.202 3.038
70 1.929 2.299 3.021 2.060 2.454 3.225 1.581 1.990 2.765 1.722 2.153 2.974
80 1.907 2.272 2.986 2.026 2.414 3.173 1.559 1.965 2.733 1.688 2.114 2.924
90 1.889 2.251 2.958 1.999 2.382 3.130 1.542 1.944 2.706 1.661 2.082 2.883
100 1.874 2.233 2.934 1.977 2.355 3.096 1.527 1.927 2.684 1.639 2.056 2.850
150 1.825 2.175 2.859 1.905 2.270 2.983 1.478 1.870 2.611 1.566 1.971 2.741
200 1.798 2.143 2.816 1.865 2.222 2.921 1.450 1.837 2.570 1.524 1.923 2.679
250 1.780 2.121 2.788 1.839 2.191 2.880 1.431 1.815 2.542 1.496 1.891 2.638
300 1.767 2.106 2.767 1.820 2.169 2.850 1.417 1.800 2.522 1.476 1.868 2.608
ⴥ 1.645 1.960 2.576 1.645 1.960 2.576 1.282 1.645 2.326 1.282 1.645 2.326

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-11

Table A.7 Critical Values for Chi-Squared Distributions 2 density curve

Shaded area = α

0 2
 α,

␯ .995 .99 .975 .95 .90 .10 .05 .025 .01 .005

1 0.000 0.000 0.001 0.004 0.016 2.706 3.843 5.025 6.637 7.882
2 0.010 0.020 0.051 0.103 0.211 4.605 5.992 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.344 12.837
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.085 16.748
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.440 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.012 18.474 20.276
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.534 20.090 21.954
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.022 21.665 23.587
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.724 26.755
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.041 19.812 22.362 24.735 27.687 29.817
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.600 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.577 32.799
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.407 7.564 8.682 10.085 24.769 27.587 30.190 33.408 35.716
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.843 7.632 8.906 10.117 11.651 27.203 30.143 32.852 36.190 38.580
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.033 8.897 10.283 11.591 13.240 29.615 32.670 35.478 38.930 41.399
22 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796
23 9.260 10.195 11.688 13.090 14.848 32.007 35.172 38.075 41.637 44.179
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.558
25 10.519 11.523 13.120 14.611 16.473 34.381 37.652 40.646 44.313 46.925
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.807 12.878 14.573 16.151 18.114 36.741 40.113 43.194 46.962 49.642
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.120 14.256 16.147 17.708 19.768 39.087 42.557 45.772 49.586 52.333
30 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
31 14.457 15.655 17.538 19.280 21.433 41.422 44.985 48.231 52.190 55.000
32 15.134 16.362 18.291 20.072 22.271 42.585 46.194 49.480 53.486 56.328
33 15.814 17.073 19.046 20.866 23.110 43.745 47.400 50.724 54.774 57.646
34 16.501 17.789 19.806 21.664 23.952 44.903 48.602 51.966 56.061 58.964
35 17.191 18.508 20.569 22.465 24.796 46.059 49.802 53.203 57.340 60.272
36 17.887 19.233 21.336 23.269 25.643 47.212 50.998 54.437 58.619 61.581
37 18.584 19.960 22.105 24.075 26.492 48.363 52.192 55.667 59.891 62.880
38 19.289 20.691 22.878 24.884 27.343 49.513 53.384 56.896 61.162 64.181
39 19.994 21.425 23.654 25.695 28.196 50.660 54.572 58.119 62.426 65.473
40 20.706 22.164 24.433 26.509 29.050 51.805 55.758 59.342 63.691 66.766

b
For v . 40, x2a,v , 2 2 3
, va1 2 1 za
9v B 9v

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-12 Appendix Tables

Table A.8 t Curve Tail Areas

t curve Area to the

right of t

0
t

t ␯ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.0 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500
0.1 .468 .465 .463 .463 .462. .462 .462 .461 .461 .461 .461 .461 .461 .461 .461 .461 .461 .461
0.2 .437 .430 .427 .426 .425 .424 .424 .423 .423 .423 .423 .422 .422 .422 .422 .422 .422 .422
0.3 .407 .396 .392 .390 .388 .387 .386 .386 .386 .385 .385 .385 .384 .384 .384 .384 .384 .384
0.4 .379 .364 .358 .355 .353 .352 .351 .350 .349 .349 .348 .348 .348 .347 .347 .347 .347 .347
0.5 .352 .333 .326 .322 .319 .317 .316 .315 .315 .314 .313 .313 .313 .312 .312 .312 .312 .312
0.6 .328 .305 .295 .290 .287 .285 .284 .283 .282 .281 .280 .280 .279 .279 .279 .278 .278 .278
0.7 .306 .278 .267 .261 .258 .255 .253 .252 .251 .250 .249 .249 .248 .247 .247 .247 .247 .246
0.8 .285 .254 .241 .234 .230 .227 .225 .223 .222 .221 .220 .220 .219 .218 .218 .218 .217 .217
0.9 .267 .232 .217 .210 .205 .201 .199 .197 .196 .195 .194 .193 .192 .191 .191 .191 .190 .190
1.0 .250 .211 .196 .187 .182 .178 .175 .173 .172 .170 .169 .169 .168 .167 .167 .166 .166 .165
1.1 .235 .193 .176 .167 .162 .157 .154 .152 .150 .149 .147 .146 .146 .144 .144 .144 .143 .143
1.2 .221 .177 .158 .148 .142 .138 .135 .132 .130 .129 .128 .127 .126 .124 .124 .124 .123 .123
1.3 .209 .162 .142 .132 .125 .121 .117 .115 .113 .111 .110 .109 .108 .107 .107 .106 .105 .105
1.4 .197 .148 .128 .117 .110 .106 .102 .100 .098 .096 .095 .093 .092 .091 .091 .090 .090 .089
1.5 .187 .136 .115 .104 .097 .092 .089 .086 .084 .082 .081 .080 .079 .077 .077 .077 .076 .075
1.6 .178 .125 .104 .092 .085 .080 .077 .074 .072 .070 .069 .068 .067 .065 .065 .065 .064 .064
1.7 .169 .116 .094 .082 .075 .070 .065 .064 .062 .060 .059 .057 .056 .055 .055 .054 .054 .053
1.8 .161 .107 .085 .073 .066 .061 .057 .055 .053 .051 .050 .049 .048 .046 .046 .045 .045 .044
1.9 .154 .099 .077 .065 .058 .053 .050 .047 .045 .043 .042 .041 .040 .038 .038 .038 .037 .037
2.0 .148 .092 .070 .058 .051 .046 .043 .040 .038 .037 .035 .034 .033 .032 .032 .031 .031 .030
2.1 .141 .085 .063 .052 .045 .040 .037 .034 .033 .031 .030 .029 .028 .027 .027 .026 .025 .025
2.2 .136 .079 .058 .046 .040 .035 .032 .029 .028 .026 .025 .024 .023 .022 .022 .021 .021 .021
2.3 .131 .074 .052 .041 .035 .031 .027 .025 .023 .022 .021 .020 .019 .018 .018 .018 .017 .017
2.4 .126 .069 .048 .037 .031 .027 .024 .022 .020 .019 .018 .017 .016 .015 .015 .014 .014 .014
2.5 .121 .065 .044 .033 .027 .023 .020 .018 .017 .016 .015 .014 .013 .012 .012 .012 .011 .011
2.6 .117 .061 .040 .030 .024 .020 .018 .016 .014 .013 .012 .012 .011 .010 .010 .010 .009 .009
2.7 .113 .057 .037 .027 .021 .018 .015 .014 .012 .011 .010 .010 .009 .008 .008 .008 .008 .007
2.8 .109 .054 .034 .024 .019 .016 .013 .012 .010 .009 .009 .008 .008 .007 .007 .006 .006 .006
2.9 .106 .051 .031 .022 .017 .014 .011 .010 .009 .008 .007 .007 .006 .005 .005 .005 .005 .005
3.0 .102 .048 .029 .020 .015 .012 .010 .009 .007 .007 .006 .006 .005 .004 .004 .004 .004 .004
3.1 .099 .045 .027 .018 .013 .011 .009 .007 .006 .006 .005 .005 .004 .004 .004 .003 .003 .003
3.2 .096 .043 .025 .016 .012 .009 .008 .006 .005 .005 .004 .004 .003 .003 .003 .003 .003 .002
3.3 .094 .040 .023 .015 .011 .008 .007 .005 .005 .004 .004 .003 .003 .002 .002 .002 .002 .002
3.4 .091 .038 .021 .014 .010 .007 .006 .005 .004 .003 .003 .003 .002 .002 .002 .002 .002 .002
3.5 .089 .036 .020 .012 .009 .006 .005 .004 .003 .003 .002 .002 .002 .002 .002 .001 .001 .001
3.6 .086 .035 .018 .011 .008 .006 .004 .004 .003 .002 .002 .002 .002 .001 .001 .001 .001 .001
3.7 .084 .033 .017 .010 .007 .005 .004 .003 .002 .002 .002 .002 .001 .001 .001 .001 .001 .001
3.8 .082 .031 .016 .010 .006 .004 .003 .003 .002 .002 .001 .001 .001 .001 .001 .001 .001 .001
3.9 .080 .030 .015 .009 .006 .004 .003 .002 .002 .001 .001 .001 .001 .001 .001 .001 .001 .001
4.0 .078 .029 .014 .008 .005 .004 .003 .002 .002 .001 .001 .001 .001 .001 .001 .001 .000 .000
(continued)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-13

Table A.8 t Curve Tail Areas (cont.)

t curve Area to the

right of t

0
t

t ␯ 19 20 21 22 23 24 25 26 27 28 29 30 35 40 60 120 `( 5 z)

0.0 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500
0.1 .461 .461 .461 .461 .461 .461 .461 .461 .461 .461 .461 .461 .460 .460 .460 .460 .460
0.2 .422 .422 .422 .422 .422 .422 .422 .422 .421 .421 .421 .421 .421 .421 .421 .421 .421
0.3 .384 .384 .384 .383 .383 .383 .383 .383 .383 .383 .383 .383 .383 .383 .383 .382 .382
0.4 .347 .347 .347 .347 .346 .346 .346 .346 .346 .346 .346 .346 .346 .346 .345 .345 .345
0.5 .311 .311 .311 .311 .311 .311 .311 .311 .311 .310 .310 .310 .310 .310 .309 .309 .309
0.6 .278 .278 .278 .277 .277 .277 .277 .277 .277 .277 .277 .277 .276 .276 .275 .275 .274
0.7 .246 .246 .246 .246 .245 .245 .245 .245 .245 .245 .245 .245 .244 .244 .243 .243 .242
0.8 .217 .217 .216 .216 .216 .216 .216 .215 .215 .215 .215 .215 .215 .214 .213 .213 .212
0.9 .190 .189 .189 .189 .189 .189 .188 .188 .188 .188 .188 .188 .187 .187 .186 .185 .184
1.0 .165 .165 .164 .164 .164 .164 .163 .163 .163 .163 .163 .163 .162 .162 .161 .160 .159
1.1 .143 .142 .142 .142 .141 .141 .141 .141 .141 .140 .140 .140 .139 .139 .138 .137 .136
1.2 .122 .122 .122 .121 .121 .121 .121 .120 .120 .120 .120 .120 .119 .119 .117 .116 .115
1.3 .105 .104 .104 .104 .103 .103 .103 .103 .102 .102 .102 .102 .101 .101 .099 .098 .097
1.4 .089 .089 .088 .088 .087 .087 .087 .087 .086 .086 .086 .086 .085 .085 .083 .082 .081
1.5 .075 .075 .074 .074 .074 .073 .073 .073 .073 .072 .072 .072 .071 .071 .069 .068 .067
1.6 .063 .063 .062 .062 .062 .061 .061 .061 .061 .060 .060 .060 .059 .059 .057 .056 .055
1.7 .053 .052 .052 .052 .051 .051 .051 .051 .050 .050 .050 .050 .049 .048 .047 .046 .045
1.8 .044 .043 .043 .043 .042 .042 .042 .042 .042 .041 .041 .041 .040 .040 .038 .037 .036
1.9 .036 .036 .036 .035 .035 .035 .035 .034 .034 .034 .034 .034 .033 .032 .031 .030 .029
2.0 .030 .030 .029 .029 .029 .028 .028 .028 .028 .028 .027 .027 .027 .026 .025 .024 .023
2.1 .025 .024 .024 .024 .023 .023 .023 .023 .023 .022 .022 .022 .022 .021 .020 .019 .018
2.2 .020 .020 .020 .019 .019 .019 .019 .018 .018 .018 .018 .018 .017 .017 .016 .015 .014
2.3 .016 .016 .016 .016 .015 .015 .015 .015 .015 .015 .014 .014 .014 .013 .012 .012 .011
2.4 .013 .013 .013 .013 .012 .012 .012 .012 .012 .012 .012 .011 .011 .011 .010 .009 .008
2.5 .011 .011 .010 .010 .010 .010 .010 .010 .009 .009 .009 .009 .009 .008 .008 .007 .006
2.6 .009 .009 .008 .008 .008 .008 .008 .008 .007 .007 .007 .007 .007 .007 .006 .005 .005
2.7 .007 .007 .007 .007 .006 .006 .006 .006 .006 .006 .006 .006 .005 .005 .004 .004 .003
2.8 .006 .006 .005 .005 .005 .005 .005 .005 .005 .005 .005 .004 .004 .004 .003 .003 .003
2.9 .005 .004 .004 .004 .004 .004 .004 .004 .004 .004 .004 .003 .003 .003 .003 .002 .002
3.0 .004 .004 .003 .003 .003 .003 .003 .003 .003 .003 .003 .003 .002 .002 .002 .002 .001
3.1 .003 .003 .003 .003 .003 .002 .002 .002 .002 .002 .002 .002 .002 .002 .001 .001 .001
3.2 .002 .002 .002 .002 .002 .002 .002 .002 .002 .002 .002 .002 .001 .001 .001 .001 .001
3.3 .002 .002 .002 .002 .002 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .000
3.4 .002 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .000 .000
3.5 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .000 .000 .000
3.6 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001 .000 .000 .000 .000 .000
3.7 .001 .001 .001 .001 .001 .001 .001 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000
3.8 .001 .001 .001 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
3.9 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
4.0 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-14 Appendix Tables

Table A.9 Critical Values for F Distributions

␯ 1 ⫽ numerator df

␣ 1 2 3 4 5 6 7 8 9

.100 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86
.050 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
1
.010 4052.20 4999.50 5403.40 5624.60 5763.60 5859.00 5928.40 5981.10 6022.50
.001 405,284 500,000 540,379 562,500 576,405 585,937 592,873 598,144 602,284
.100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38
.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
2
.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39
.001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39
.100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24
.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
3
.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35
.001 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86
.100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94
.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
4
.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47
.100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32
.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
5
.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24
␯2 ⴝ denominator df

.100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96
.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
6
.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98
.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 18.69
.100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72
.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
7
.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72
.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.33
.100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56
.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
8
.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91
.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.77
.100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
9
.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
.100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35
.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
10
.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.96
.100 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27
.050 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
11
.010 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63
.001 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12
.100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21
.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
12
.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39
.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48
(continued)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-15

Table A.9 Critical Values for F Distributions (cont.)

␯1 ⫽ numerator df

10 12 15 20 25 30 40 50 60 120 1000

60.19 60.71 61.22 61.74 62.05 62.26 62.53 62.69 62.79 63.06 63.30
241.88 243.91 245.95 248.01 249.26 250.10 251.14 251.77 252.20 253.25 254.19
6055.80 6106.30 6157.30 6208.70 6239.80 6260.60 6286.80 6302.50 6313.00 6339.40 6362.70
605,621 610,668 615,764 620,908 624,017 626,099 628,712 630,285 631,337 633,972 636,301
9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.47 9.48 9.49
19.40 19.41 19.43 19.45 19.46 19.46 19.47 19.48 19.48 19.49 19.49
99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.48 99.49 99.50
999.40 999.42 999.43 999.45 999.46 999.47 999.47 999.48 999.48 999.49 999.50
5.23 5.22 5.20 5.18 5.17 5.17 5.16 5.15 5.15 5.14 5.13
8.79 8.74 8.70 8.66 8.63 8.62 8.59 8.58 8.57 8.55 8.53
27.23 27.05 26.87 26.69 26.58 26.50 26.41 26.35 26.32 26.22 26.14
129.25 128.32 127.37 126.42 125.84 125.45 124.96 124.66 124.47 123.97 123.53
3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.80 3.79 3.78 3.76
5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.70 5.69 5.66 5.63
14.55 14.37 14.20 14.02 13.91 13.84 13.75 13.69 13.65 13.56 13.47
48.05 47.41 46.76 46.10 45.70 45.43 45.09 44.88 44.75 44.40 44.09
3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.15 3.14 3.12 3.11
4.74 4.68 4.62 4.56 4.52 4.50 4.46 4.44 4.43 4.40 4.37
10.05 9.89 9.72 9.55 9.45 9.38 9.29 9.24 9.20 9.11 9.03
26.92 26.42 25.91 25.39 25.08 24.87 24.60 24.44 24.33 24.06 23.82
2.94 2.90 2.87 2.84 2.81 2.80 2.78 2.77 2.76 2.74 2.72
4.06 4.00 3.94 3.87 3.83 3.81 3.77 3.75 3.74 3.70 3.67
7.87 7.72 7.56 7.40 7.30 7.23 7.14 7.09 7.06 6.97 6.89
18.41 17.99 17.56 17.12 16.85 16.67 16.44 16.31 16.21 15.98 15.77
2.70 2.67 2.63 2.59 2.57 2.56 2.54 2.52 2.51 2.49 2.47
3.64 3.57 3.51 3.44 3.40 3.38 3.34 3.32 3.30 3.27 3.23
6.62 6.47 6.31 6.16 6.06 5.99 5.91 5.86 5.82 5.74 5.66
14.08 13.71 13.32 12.93 12.69 12.53 12.33 12.20 12.12 11.91 11.72
2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.35 2.34 2.32 2.30
3.35 3.28 3.22 3.15 3.11 3.08 3.04 3.02 3.01 2.97 2.93
5.81 5.67 5.52 5.36 5.26 5.20 5.12 5.07 5.03 4.95 4.87
11.54 11.19 10.84 10.48 10.26 10.11 9.92 9.80 9.73 9.53 9.36
2.42 2.38 2.34 2.30 2.27 2.25 2.23 2.22 2.21 2.18 2.16
3.14 3.07 3.01 2.94 2.89 2.86 2.83 2.80 2.79 2.75 2.71
5.26 5.11 4.96 4.81 4.71 4.65 4.57 4.52 4.48 4.40 4.32
9.89 9.57 9.24 8.90 8.69 8.55 8.37 8.26 8.19 8.00 7.84
2.32 2.28 2.24 2.20 2.17 2.16 2.13 2.12 2.11 2.08 2.06
2.98 2.91 2.85 2.77 2.73 2.70 2.66 2.64 2.62 2.58 2.54
4.85 4.71 4.56 4.41 4.31 4.25 4.17 4.12 4.08 4.00 3.92
8.75 8.45 8.13 7.80 7.60 7.47 7.30 7.19 7.12 6.94 6.78
2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.04 2.03 2.00 1.98
2.85 2.79 2.72 2.65 2.60 2.57 2.53 2.51 2.49 2.45 2.41
4.54 4.40 4.25 4.10 4.01 3.94 3.86 3.81 3.78 3.69 3.61
7.92 7.63 7.32 7.01 6.81 6.68 6.52 6.42 6.35 6.18 6.02
2.19 2.15 2.10 2.06 2.03 2.01 1.99 1.97 1.96 1.93 1.91
2.75 2.69 2.62 2.54 2.50 2.47 2.43 2.40 2.38 2.34 2.30
4.30 4.16 4.01 3.86 3.76 3.70 3.62 3.57 3.54 3.45 3.37
7.29 7.00 6.71 6.40 6.22 6.09 5.93 5.83 5.76 5.59 5.44
(continued)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-16 Appendix Tables

Table A.9 Critical Values for F Distributions (cont.)

␯1 ⴝ numerator df

␣ 1 2 3 4 5 6 7 8 9

.100 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16
.050 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
13
.010 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19
.001 17.82 12.31 10.21 9.07 8.35 7.86 7.49 7.21 6.98
.100 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12
.050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
14
.010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
.001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.58
.100 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09
.050 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
15
.010 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
.001 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26
.100 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06
.050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
16
.010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78
.001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.98
.100 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03
.050 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
17
.010 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68
.001 15.72 10.66 8.73 7.68 7.02 6.56 6.22 5.96 5.75
.100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00
␯2 ⫽ denominator df

.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
18
.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60
.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56
.100 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98
.050 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
19
.010 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52
.001 15.08 10.16 8.28 7.27 6.62 6.18 5.85 5.59 5.39
.100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96
.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
20
.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24
.100 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95
.050 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
21
.010 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40
.001 14.59 9.77 7.94 6.95 6.32 5.88 5.56 5.31 5.11
.100 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93
.050 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
22
.010 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35
.001 14.38 9.61 7.80 6.81 6.19 5.76 5.44 5.19 4.99
.100 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92
.050 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32
23
.010 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30
.001 14.20 9.47 7.67 6.70 6.08 5.65 5.33 5.09 4.89
.100 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91
.050 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
24
.010 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
.001 14.03 9.34 7.55 6.59 5.98 5.55 5.23 4.99 4.80
(continued)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-17

Table A.9 Critical Values for F Distributions (cont.)

␯1 ⫽ numerator df

10 12 15 20 25 30 40 50 60 120 1000

2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.92 1.90 1.88 1.85
2.67 2.60 2.53 2.46 2.41 2.38 2.34 2.31 2.30 2.25 2.21
4.10 3.96 3.82 3.66 3.57 3.51 3.43 3.38 3.34 3.25 3.18
6.80 6.52 6.23 5.93 5.75 5.63 5.47 5.37 5.30 5.14 4.99
2.10 2.05 2.01 1.96 1.93 1.91 1.89 1.87 1.86 1.83 1.80
2.60 2.53 2.46 2.39 2.34 2.31 2.27 2.24 2.22 2.18 2.14
3.94 3.80 3.66 3.51 3.41 3.35 3.27 3.22 3.18 3.09 3.02
6.40 6.13 5.85 5.56 5.38 5.25 5.10 5.00 4.94 4.77 4.62
2.06 2.02 1.97 1.92 1.89 1.87 1.85 1.83 1.82 1.79 1.76
2.54 2.48 2.40 2.33 2.28 2.25 2.20 2.18 2.16 2.11 2.07
3.80 3.67 3.52 3.37 3.28 3.21 3.13 3.08 3.05 2.96 2.88
6.08 5.81 5.54 5.25 5.07 4.95 4.80 4.70 4.64 4.47 4.33
2.03 1.99 1.94 1.89 1.86 1.84 1.81 1.79 1.78 1.75 1.72
2.49 2.42 2.35 2.28 2.23 2.19 2.15 2.12 2.11 2.06 2.02
3.69 3.55 3.41 3.26 3.16 3.10 3.02 2.97 2.93 2.84 2.76
5.81 5.55 5.27 4.99 4.82 4.70 4.54 4.45 4.39 4.23 4.08
2.00 1.96 1.91 1.86 1.83 1.81 1.78 1.76 1.75 1.72 1.69
2.45 2.38 2.31 2.23 2.18 2.15 2.10 2.08 2.06 2.01 1.97
3.59 3.46 3.31 3.16 3.07 3.00 2.92 2.87 2.83 2.75 2.66
5.58 5.32 5.05 4.78 4.60 4.48 4.33 4.24 4.18 4.02 3.87
1.98 1.93 1.89 1.84 1.80 1.78 1.75 1.74 1.72 1.69 1.66
2.41 2.34 2.27 2.19 2.14 2.11 2.06 2.04 2.02 1.97 1.92
3.51 3.37 3.23 3.08 2.98 2.92 2.84 2.78 2.75 2.66 2.58
5.39 5.13 4.87 4.59 4.42 4.30 4.15 4.06 4.00 3.84 3.69
1.96 1.91 1.86 1.81 1.78 1.76 1.73 1.71 1.70 1.67 1.64
2.38 2.31 2.23 2.16 2.11 2.07 2.03 2.00 1.98 1.93 1.88
3.43 3.30 3.15 3.00 2.91 2.84 2.76 2.71 2.67 2.58 2.50
5.22 4.97 4.70 4.43 4.26 4.14 3.99 3.90 3.84 3.68 3.53
1.94 1.89 1.84 1.79 1.76 1.74 1.71 1.69 1.68 1.64 1.61
2.35 2.28 2.20 2.12 2.07 2.04 1.99 1.97 1.95 1.90 1.85
3.37 3.23 3.09 2.94 2.84 2.78 2.69 2.64 2.61 2.52 2.43
5.08 4.82 4.56 4.29 4.12 4.00 3.86 3.77 3.70 3.54 3.40
1.92 1.87 1.83 1.78 1.74 1.72 1.69 1.67 1.66 1.62 1.59
2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.94 1.92 1.87 1.82
3.31 3.17 3.03 2.88 2.79 2.72 2.64 2.58 2.55 2.46 2.37
4.95 4.70 4.44 4.17 4.00 3.88 3.74 3.64 3.58 3.42 3.28
1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.65 1.64 1.60 1.57
2.30 2.23 2.15 2.07 2.02 1.98 1.94 1.91 1.89 1.84 1.79
3.26 3.12 2.98 2.83 2.73 2.67 2.58 2.53 2.50 2.40 2.32
4.83 4.58 4.33 4.06 3.89 3.78 3.63 3.54 3.48 3.32 3.17
1.89 1.84 1.80 1.74 1.71 1.69 1.66 1.64 1.62 1.59 1.55
2.27 2.20 2.13 2.05 2.00 1.96 1.91 1.88 1.86 1.81 1.76
3.21 3.07 2.93 2.78 2.69 2.62 2.54 2.48 2.45 2.35 2.27
4.73 4.48 4.23 3.96 3.79 3.68 3.53 3.44 3.38 3.22 3.08
1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.62 1.61 1.57 1.54
2.25 2.18 2.11 2.03 1.97 1.94 1.89 1.86 1.84 1.79 1.74
3.17 3.03 2.89 2.74 2.64 2.58 2.49 2.44 2.40 2.31 2.22
4.64 4.39 4.14 3.87 3.71 3.59 3.45 3.36 3.29 3.14 2.99
(continued)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-18 Appendix Tables

Table A.9 Critical Values for F Distributions (cont.)

␯1 ⫽ numerator df

␣ 1 2 3 4 5 6 7 8 9

.100 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89
.050 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
25
.010 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
.001 13.88 9.22 7.45 6.49 5.89 5.46 5.15 4.91 4.71
.100 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88
.050 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27
26
.010 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18
.001 13.74 9.12 7.36 6.41 5.80 5.38 5.07 4.83 4.64
.100 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87
.050 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
27
.010 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15
.001 13.61 9.02 7.27 6.33 5.73 5.31 5.00 4.76 4.57
.100 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87
.050 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24
28
.010 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12
.001 13.50 8.93 7.19 6.25 5.66 5.24 4.93 4.69 4.50
.100 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86
.050 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
29
.010 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
.001 13.39 8.85 7.12 6.19 5.59 5.18 4.87 4.64 4.45
␯2 ⴝ denominator df

.100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85
.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
30
.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39
.100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79
.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12
40
.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89
.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 4.02
.100 2.81 2.41 2.20 2.06 1.97 1.90 1.84 1.80 1.76
.050 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07
50
.010 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78
.001 12.22 7.96 6.34 5.46 4.90 4.51 4.22 4.00 3.82
.100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74
.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
60
.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.69
.100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69
.050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97
100
.010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59
.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.44
.100 2.73 2.33 2.11 1.97 1.88 1.80 1.75 1.70 1.66
.050 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93
200
.010 6.76 4.71 3.88 3.41 3.11 2.89 2.73 2.60 2.50
.001 11.15 7.15 5.63 4.81 4.29 3.92 3.65 3.43 3.26
.100 2.71 2.31 2.09 1.95 1.85 1.78 1.72 1.68 1.64
.050 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89
1000
.010 6.66 4.63 3.80 3.34 3.04 2.82 2.66 2.53 2.43
.001 10.89 6.96 5.46 4.65 4.14 3.78 3.51 3.30 3.13
(continued)

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 Appendix Tables A-19

Table A.9 Critical Values for F Distributions (cont.)

␯1 ⫽ numerator df

10 12 15 20 25 30 40 50 60 120 1000

1.87 1.82 1.77 1.72 1.68 1.66 1.63 1.61 1.59 1.56 1.52
2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.84 1.82 1.77 1.72
3.13 2.99 2.85 2.70 2.60 2.54 2.45 2.40 2.36 2.27 2.18
4.56 4.31 4.06 3.79 3.63 3.52 3.37 3.28 3.22 3.06 2.91
1.86 1.81 1.76 1.71 1.67 1.65 1.61 1.59 1.58 1.54 1.51
2.22 2.15 2.07 1.99 1.94 1.90 1.85 1.82 1.80 1.75 1.70
3.09 2.96 2.81 2.66 2.57 2.50 2.42 2.36 2.33 2.23 2.14
4.48 4.24 3.99 3.72 3.56 3.44 3.30 3.21 3.15 2.99 2.84
1.85 1.80 1.75 1.70 1.66 1.64 1.60 1.58 1.57 1.53 1.50
2.20 2.13 2.06 1.97 1.92 1.88 1.84 1.81 1.79 1.73 1.68
3.06 2.93 2.78 2.63 2.54 2.47 2.38 2.33 2.29 2.20 2.11
4.41 4.17 3.92 3.66 3.49 3.38 3.23 3.14 3.08 2.92 2.78
1.84 1.79 1.74 1.69 1.65 1.63 1.59 1.57 1.56 1.52 1.48
2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.79 1.77 1.71 1.66
3.03 2.90 2.75 2.60 2.51 2.44 2.35 2.30 2.26 2.17 2.08
4.35 4.11 3.86 3.60 3.43 3.32 3.18 3.09 3.02 2.86 2.72
1.83 1.78 1.73 1.68 1.64 1.62 1.58 1.56 1.55 1.51 1.47
2.18 2.10 2.03 1.94 1.89 1.85 1.81 1.77 1.75 1.70 1.65
3.00 2.87 2.73 2.57 2.48 2.41 2.33 2.27 2.23 2.14 2.05
4.29 4.05 3.80 3.54 3.38 3.27 3.12 3.03 2.97 2.81 2.66
1.82 1.77 1.72 1.67 1.63 1.61 1.57 1.55 1.54 1.50 1.46
2.16 2.09 2.01 1.93 1.88 1.84 1.79 1.76 1.74 1.68 1.63
2.98 2.84 2.70 2.55 2.45 2.39 2.30 2.25 2.21 2.11 2.02
4.24 4.00 3.75 3.49 3.33 3.22 3.07 2.98 2.92 2.76 2.61
1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.48 1.47 1.42 1.38
2.08 2.00 1.92 1.84 1.78 1.74 1.69 1.66 1.64 1.58 1.52
2.80 2.66 2.52 2.37 2.27 2.20 2.11 2.06 2.02 1.92 1.82
3.87 3.64 3.40 3.14 2.98 2.87 2.73 2.64 2.57 2.41 2.25
1.73 1.68 1.63 1.57 1.53 1.50 1.46 1.44 1.42 1.38 1.33
2.03 1.95 1.87 1.78 1.73 1.69 1.63 1.60 1.58 1.51 1.45
2.70 2.56 2.42 2.27 2.17 2.10 2.01 1.95 1.91 1.80 1.70
3.67 3.44 3.20 2.95 2.79 2.68 2.53 2.44 2.38 2.21 2.05
1.71 1.66 1.60 1.54 1.50 1.48 1.44 1.41 1.40 1.35 1.30
1.99 1.92 1.84 1.75 1.69 1.65 1.59 1.56 1.53 1.47 1.40
2.63 2.50 2.35 2.20 2.10 2.03 1.94 1.88 1.84 1.73 1.62
3.54 3.32 3.08 2.83 2.67 2.55 2.41 2.32 2.25 2.08 1.92
1.66 1.61 1.56 1.49 1.45 1.42 1.38 1.35 1.34 1.28 1.22
1.93 1.85 1.77 1.68 1.62 1.57 1.52 1.48 1.45 1.38 1.30
2.50 2.37 2.22 2.07 1.97 1.89 1.80 1.74 1.69 1.57 1.45
3.30 3.07 2.84 2.59 2.43 2.32 2.17 2.08 2.01 1.83 1.64
1.63 1.58 1.52 1.46 1.41 1.38 1.34 1.31 1.29 1.23 1.16
1.88 1.80 1.72 1.62 1.56 1.52 1.46 1.41 1.39 1.30 1.21
2.41 2.27 2.13 1.97 1.87 1.79 1.69 1.63 1.58 1.45 1.30
3.12 2.90 2.67 2.42 2.26 2.15 2.00 1.90 1.83 1.64 1.43
1.61 1.55 1.49 1.43 1.38 1.35 1.30 1.27 1.25 1.18 1.08
1.84 1.76 1.68 1.58 1.52 1.47 1.41 1.36 1.33 1.24 1.11
2.34 2.20 2.06 1.90 1.79 1.72 1.61 1.54 1.50 1.35 1.16
2.99 2.77 2.54 2.30 2.14 2.02 1.87 1.77 1.69 1.49 1.22

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-20 Appendix Tables

Table A.10 Critical Values for Studentized Range Distributions

␯ ␣ 2 3 4 5 6 7 8 9 10 11 12

5 .05 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17 7.32
.01 5.70 6.98 7.80 8.42 8.91 9.32 9.67 9.97 10.24 10.48 10.70
6 .05 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65 6.79
.01 5.24 6.33 7.03 7.56 7.97 8.32 8.61 8.87 9.10 9.30 9.48
7 .05 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30 6.43
.01 4.95 5.92 6.54 7.01 7.37 7.68 7.94 8.17 8.37 8.55 8.71
8 .05 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05 6.18
.01 4.75 5.64 6.20 6.62 6.96 7.24 7.47 7.68 7.86 8.03 8.18
9 .05 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87 5.98
.01 4.60 5.43 5.96 6.35 6.66 6.91 7.13 7.33 7.49 7.65 7.78
10 .05 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72 5.83
.01 4.48 5.27 5.77 6.14 6.43 6.67 6.87 7.05 7.21 7.36 7.49
11 .05 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61 5.71
.01 4.39 5.15 5.62 5.97 6.25 6.48 6.67 6.84 6.99 7.13 7.25
12 .05 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51 5.61
.01 4.32 5.05 5.50 5.84 6.10 6.32 6.51 6.67 6.81 6.94 7.06
13 .05 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43 5.53
.01 4.26 4.96 5.40 5.73 5.98 6.19 6.37 6.53 6.67 6.79 6.90
14 .05 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36 5.46
.01 4.21 4.89 5.32 5.63 5.88 6.08 6.26 6.41 6.54 6.66 6.77
15 .05 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31 5.40
.01 4.17 4.84 5.25 5.56 5.80 5.99 6.16 6.31 6.44 6.55 6.66
16 .05 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26 5.35
.01 4.13 4.79 5.19 5.49 5.72 5.92 6.08 6.22 6.35 6.46 6.56
17 .05 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21 5.31
.01 4.10 4.74 5.14 5.43 5.66 5.85 6.01 6.15 6.27 6.38 6.48
18 .05 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17 5.27
.01 4.07 4.70 5.09 5.38 5.60 5.79 5.94 6.08 6.20 6.31 6.41
19 .05 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14 5.23
.01 4.05 4.67 5.05 5.33 5.55 5.73 5.89 6.02 6.14 6.25 6.34
20 .05 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11 5.20
.01 4.02 4.64 5.02 5.29 5.51 5.69 5.84 5.97 6.09 6.19 6.28
24 .05 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01 5.10
.01 3.96 4.55 4.91 5.17 5.37 5.54 5.69 5.81 5.92 6.02 6.11
30 .05 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92 5.00
.01 3.89 4.45 4.80 5.05 5.24 5.40 5.54 5.65 5.76 5.85 5.93
40 .05 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82 4.90
.01 3.82 4.37 4.70 4.93 5.11 5.26 5.39 5.50 5.60 5.69 5.76
60 .05 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73 4.81
.01 3.76 4.28 4.59 4.82 4.99 5.13 5.25 5.36 5.45 5.53 5.60
120 .05 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64 4.71
.01 3.70 4.20 4.50 4.71 4.87 5.01 5.12 5.21 5.30 5.37 5.44
∞ .05 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62
.01 3.64 4.12 4.40 4.60 4.76 4.88 4.99 5.08 5.16 5.23 5.29

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 Appendix Tables A-21

Table A.11 Chi-Squared Curve Tail Areas

Upper-tail Area ␯⫽1 ␯⫽2 ␯⫽3 ␯⫽4 ␯⫽5

. .100 , 2.70 , 4.60 , 6.25 , 7.77 ⬍ 9.23

.100 2.70 4.60 6.25 7.77 9.23
.095 2.78 4.70 6.36 7.90 9.37
.090 2.87 4.81 6.49 8.04 9.52
.085 2.96 4.93 6.62 8.18 9.67
.080 3.06 5.05 6.75 8.33 9.83
.075 3.17 5.18 6.90 8.49 10.00
.070 3.28 5.31 7.06 8.66 10.19
.065 3.40 5.46 7.22 8.84 10.38
.060 3.53 5.62 7.40 9.04 10.59
.055 3.68 5.80 7.60 9.25 10.82
.050 3.84 5.99 7.81 9.48 11.07
.045 4.01 6.20 8.04 9.74 11.34
.040 4.21 6.43 8.31 10.02 11.64
.035 4.44 6.70 8.60 10.34 11.98
.030 4.70 7.01 8.94 10.71 12.37
.025 5.02 7.37 9.34 11.14 12.83
.020 5.41 7.82 9.83 11.66 13.38
.015 5.91 8.39 10.46 12.33 14.09
.010 6.63 9.21 11.34 13.27 15.08
.005 7.87 10.59 12.83 14.86 16.74
.001 10.82 13.81 16.26 18.46 20.51
⬍ .001 ⬎ 10.82 ⬎ 13.81 ⬎ 16.26 ⬎ 18.46 ⬎ 20.51
Upper-tail Area ␯⫽6 ␯⫽7 ␯⫽8 ␯⫽9 ␯ ⫽ 10

⬎ .100 ⬍ 10.64 ⬍ 12.01 ⬍ 13.36 ⬍ 14.68 ⬍ 15.98

.100 10.64 12.01 13.36 14.68 15.98
.095 10.79 12.17 13.52 14.85 16.16
.090 10.94 12.33 13.69 15.03 16.35
.085 11.11 12.50 13.87 15.22 16.54
.080 11.28 12.69 14.06 15.42 16.75
.075 11.46 12.88 14.26 15.63 16.97
.070 11.65 13.08 14.48 15.85 17.20
.065 11.86 13.30 14.71 16.09 17.44
.060 12.08 13.53 14.95 16.34 17.71
.055 12.33 13.79 15.22 16.62 17.99
.050 12.59 14.06 15.50 16.91 18.30
.045 12.87 14.36 15.82 17.24 18.64
.040 13.19 14.70 16.17 17.60 19.02
.035 13.55 15.07 16.56 18.01 19.44
.030 13.96 15.50 17.01 18.47 19.92
.025 14.44 16.01 17.53 19.02 20.48
.020 15.03 16.62 18.16 19.67 21.16
.015 15.77 17.39 18.97 20.51 22.02
.010 16.81 18.47 20.09 21.66 23.20
.005 18.54 20.27 21.95 23.58 25.18
.001 22.45 24.32 26.12 27.87 29.58
⬍ .001 ⬎ 22.45 ⬎ 24.32 ⬎ 26.12 ⬎ 27.87 ⬎ 29.58
(continued)

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-22 Appendix Tables

Table A.11 Chi-Squared Curve Tail Areas (cont.)

Upper-tail Area ␯ ⫽ 11 ␯ ⫽ 12 ␯ ⫽ 13 ␯ ⫽ 14 ␯ ⫽ 15

⬎ .100 ⬍ 17.27 ⬍ 18.54 ⬍ 19.81 ⬍ 21.06 ⬍ 22.30

.100 17.27 18.54 19.81 21.06 22.30
.095 17.45 18.74 20.00 21.26 22.51
.090 17.65 18.93 20.21 21.47 22.73
.085 17.85 19.14 20.42 21.69 22.95
.080 18.06 19.36 20.65 21.93 23.19
.075 18.29 19.60 20.89 22.17 23.45
.070 18.53 19.84 21.15 22.44 23.72
.065 18.78 20.11 21.42 22.71 24.00
.060 19.06 20.39 21.71 23.01 24.31
.055 19.35 20.69 22.02 23.33 24.63
.050 19.67 21.02 22.36 23.68 24.99
.045 20.02 21.38 22.73 24.06 25.38
.040 20.41 21.78 23.14 24.48 25.81
.035 20.84 22.23 23.60 24.95 26.29
.030 21.34 22.74 24.12 25.49 26.84
.025 21.92 23.33 24.73 26.11 27.48
.020 22.61 24.05 25.47 26.87 28.25
.015 23.50 24.96 26.40 27.82 29.23
.010 24.72 26.21 27.68 29.14 30.57
.005 26.75 28.29 29.81 31.31 32.80
.001 31.26 32.90 34.52 36.12 37.69
⬍ .001 ⬎ 31.26 ⬎ 32.90 ⬎ 34.52 ⬎ 36.12 ⬎ 37.69

Upper-tail Area ␯ ⫽ 16 ␯ ⫽ 17 ␯ ⫽ 18 ␯ ⫽ 19 ␯ ⫽ 20

⬎ .100 ⬍ 23.54 ⬍ 24.77 ⬍ 25.98 ⬍ 27.20 ⬍ 28.41

.100 23.54 24.76 25.98 27.20 28.41
.095 23.75 24.98 26.21 27.43 28.64
.090 23.97 25.21 26.44 27.66 28.88
.085 24.21 25.45 26.68 27.91 29.14
.080 24.45 25.70 26.94 28.18 29.40
.075 24.71 25.97 27.21 28.45 29.69
.070 24.99 26.25 27.50 28.75 29.99
.065 25.28 26.55 27.81 29.06 30.30
.060 25.59 26.87 28.13 29.39 30.64
.055 25.93 27.21 28.48 29.75 31.01
.050 26.29 27.58 28.86 30.14 31.41
.045 26.69 27.99 29.28 30.56 31.84
.040 27.13 28.44 29.74 31.03 32.32
.035 27.62 28.94 30.25 31.56 32.85
.030 28.19 29.52 30.84 32.15 33.46
.025 28.84 30.19 31.52 32.85 34.16
.020 29.63 30.99 32.34 33.68 35.01
.015 30.62 32.01 33.38 34.74 36.09
.010 32.00 33.40 34.80 36.19 37.56
.005 34.26 35.71 37.15 38.58 39.99
.001 39.25 40.78 42.31 43.81 45.31
⬍ .001 ⬎ 39.25 ⬎ 40.78 ⬎ 42.31 ⬎ 43.81 ⬎ 45.31

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 Appendix Tables A-23

Table A.12 Critical Values for the Ryan-Joiner Test of Normality

.10 .05 .01

5 .9033 .8804 .8320

10 .9347 .9180 .8804
15 .9506 .9383 .9110
20 .9600 .9503 .9290
25 .9662 .9582 .9408
n
30 .9707 .9639 .9490
40 .9767 .9715 .9597
50 .9807 .9764 .9664
60 .9835 .9799 .9710
75 .9865 .9835 .9757

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
 A-24 Appendix Tables

Table A.13 Critical Values for the Wilcoxon Signed-Rank Test P0(S1 $ c1) 5 P(S1 $ c1 when H0 is true)

n c1 P0(S⫹ ⱖ c1) n c1 P0(S⫹ ⱖ c1)

3 6 .125 78 .011
4 9 .125 79 .009
10 .062 81 .005
5 13 .094 14 73 .108
14 .062 74 .097
15 .031 79 .052
6 17 .109 84 .025
19 .047 89 .010
20 .031 92 .005
21 .016 15 83 .104
7 22 .109 84 .094
24 .055 89 .053
26 .023 90 .047
28 .008 95 .024
8 28 .098 100 .011
30 .055 101 .009
32 .027 104 .005
34 .012 16 93 .106
35 .008 94 .096
36 .004 100 .052
9 34 .102 106 .025
37 .049 112 .011
39 .027 113 .009
42 .010 116 .005
44 .004 17 104 .103
10 41 .097 105 .095
44 .053 112 .049
47 .024 118 .025
50 .010 125 .010
52 .005 129 .005
11 48 .103 18 116 .098
52 .051 124 .049
55 .027 131 .024
59 .009 138 .010
61 .005 143 .005
12 56 .102 19 128 .098
60 .055 136 .052
61 .046 137 .048
64 .026 144 .025
68 .010 152 .010
71 .005 157 .005
13 64 .108 20 140 .101
65 .095 150 .049
69 .055 158 .024
70 .047 167 .010
74 .024 172 .005

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 Appendix Tables A-25

Table A.14 Critical Values for the Wilcoxon Rank-Sum Test P0(W $ c) 5 P(W $ c when H0 is true)

m n c P0(W ⱖ c) m n c P0(W ⱖ c)

3 3 15 .05 40 .004
4 17 .057 6 40 .041
18 .029 41 .026
5 20 .036 43 .009
21 .018 44 .004
6 22 .048 7 43 .053
23 .024 45 .024
24 .012 47 .009
7 24 .058 48 .005
26 .017 8 47 .047
27 .008 49 .023
8 27 .042 51 .009
28 .024 52 .005
29 .012 6 6 50 .047
30 .006 52 .021
4 4 24 .057 54 .008
25 .029 55 .004
26 .014 7 54 .051
5 27 .056 56 .026
28 .032 58 .011
29 .016 60 .004
30 .008 8 58 .054
6 30 .057 61 .021
32 .019 63 .01
33 .010 65 .004
34 .005 7 7 66 .049
7 33 .055 68 .027
35 .021 71 .009
36 .012 72 .006
37 .006 8 71 .047
8 36 .055 73 .027
38 .024 76 .01
40 .008 78 .005
41 .004 8 8 84 .052
5 5 36 .048 87 .025
37 .028 90 .01
39 .008 92 .005

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 A-26 Appendix Tables

Table A.15 Critical Values for the Wilcoxon Signed-Rank Interval (x(n(n11)/22c11), x(c))

Confidence Confidence Confidence

n Level (%) c n Level (%) c n Level (%) c

5 93.8 15 13 99.0 81 20 99.1 173

87.5 14 95.2 74 95.2 158
6 96.9 21 90.6 70 90.3 150
93.7 20 14 99.1 93 21 99.0 188
90.6 19 95.1 84 95.0 172
7 98.4 28 89.6 79 89.7 163
95.3 26 15 99.0 104 22 99.0 204
89.1 24 95.2 95 95.0 187
8 99.2 36 90.5 90 90.2 178
94.5 32 16 99.1 117 23 99.0 221
89.1 30 94.9 106 95.2 203
9 99.2 44 89.5 100 90.2 193
94.5 39 17 99.1 130 24 99.0 239
90.2 37 94.9 118 95.1 219
10 99.0 52 90.2 112 89.9 208
95.1 47 18 99.0 143 25 99.0 257
89.5 44 95.2 131 95.2 236
11 99.0 61 90.1 124 89.9 224
94.6 55 19 99.1 158
89.8 52 95.1 144
12 99.1 71 90.4 137
94.8 64
90.8 61

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 Appendix Tables A-27

Table A.16 Critical Values for the Wilcoxon Rank-Sum Interval (dij(mn2c11), dij(c))

Smaller Sample Size

5 6 7 8

Larger Confidence Confidence Confidence Confidence

Sample Size Level (%) c Level (%) c Level (%) c Level (%) c

5 99.2 25
94.4 22
90.5 21
6 99.1 29 99.1 34
94.8 26 95.9 31
91.8 25 90.7 29
7 99.0 33 99.2 39 98.9 44
95.2 30 94.9 35 94.7 40
89.4 28 89.9 33 90.3 38
8 98.9 37 99.2 44 99.1 50 99.0 56
95.5 34 95.7 40 94.6 45 95.0 51
90.7 32 89.2 37 90.6 43 89.5 48
9 98.8 41 99.2 49 99.2 56 98.9 62
95.8 38 95.0 44 94.5 50 95.4 57
88.8 35 91.2 42 90.9 48 90.7 54
10 99.2 46 98.9 53 99.0 61 99.1 69
94.5 41 94.4 48 94.5 55 94.5 62
90.1 39 90.7 46 89.1 52 89.9 59
11 99.1 50 99.0 58 98.9 66 99.1 75
94.8 45 95.2 53 95.6 61 94.9 68
91.0 43 90.2 50 89.6 57 90.9 65
12 99.1 54 99.0 63 99.0 72 99.0 81
95.2 49 94.7 57 95.5 66 95.3 74
89.6 46 89.8 54 90.0 62 90.2 70

Smaller Sample Size

9 10 11 12

Larger Confidence Confidence Confidence Confidence

Sample Size Level (%) c Level (%) c Level (%) c Level (%) c

9 98.9 69
95.0 63
90.6 60
10 99.0 76 99.1 84
94.7 69 94.8 76
90.5 66 89.5 72
11 99.0 83 99.0 91 98.9 99
95.4 76 94.9 83 95.3 91
90.5 72 90.1 79 89.9 86
12 99.1 90 99.1 99 99.1 108 99.0 116
95.1 82 95.0 90 94.9 98 94.8 106
90.5 78 90.7 86 89.6 93 89.9 101

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.