Enrolment No.
/Seat No_______________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER–III (NEW) EXAMINATION – SUMMER 2024
Subject Code:3130006 Date:16-07-2024
Subject Name: Probability and Statistics
Time:10:30 AM TO 01:00 PM Total Marks:70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.
4. Simple and non-programmable scientific calculators are allowed.
MARKS
Q.1 (a) Define and give the example of Random variable. 03
(b) What is the probability that a leap year selected at random will have 53 Sundays? 04
(c) In a bolt factory, three machines A, B and C produce 25%, 35% and 40% of total 07
output respectively. It was found that 5%,4% and 2% are defective bolts in the
production by machines A, B, C respectively. A bolt is chosen at random from the
total output and is found to be defective. Find the probability that it is manufactured
from (i) Machine A(ii) Machine B (iii) Machine C.
Q.2 (a) A bag contains 3 red and 4 white balls. Two draws are made without replacement. 03
What is the probability that both balls are red.
(b) The probability that a student A solves a mathematics problem is 2 04
and the
5
2
probability that a student 𝐵 solves it is 3. What is the probability that (i) the
problem is not solved (ii) both 𝐴 and 𝐵 can solve the problem, working
independently of each other?
(c) Verify that the following function 𝐹(𝑥) is a distribution function. 07
0; 𝑥<0
𝐹(𝑥) = { −
𝑥
1 − 𝑒 4; 𝑥 ≥ 0
Also, find the probabilities 𝑃(𝑋 ≤ 4), 𝑃(𝑋 ≥ 8), 𝑃(4 ≤ 𝑋 ≤ 8).
OR
(c) The probability mass function of a random variable X is zero except at the points 07
𝑋 = 0, 1, 2. At these points, it has the values
𝑃(𝑋 = 0) = 3𝑐 3 , 𝑃(𝑋 = 1) = 4𝑐 − 10𝑐 2 , 𝑃(𝑋 = 2) = 5𝑐 − 1. Find (i) 𝑐 (ii)
𝑃(𝑋 < 1) (iii) 𝑃(1 < 𝑋 ≤ 2) (iv) 𝑃(0 < 𝑋 ≤ 2).
Q.3 (a) Find the constant 𝑘 such that the function 03
𝑘𝑥 2 ; 0 < 𝑥 < 3
𝑓(𝑥) = {
0; 𝑜𝑡ℎ𝑒𝑤𝑖𝑠𝑒
is a probability density function,
(b) A random variable X has the following distribution: 04
𝑋 1 2 3 4 5 6
𝑃(𝑋 1 3 5 7 9 11
= 𝑥) 36 36 36 36 36 36
Find (i) mean (ii) variance.
(c) Compute Karl Pearson’s coefficient of correlation between 𝑋 and 𝑌 for the 07
following data:
𝑥 10 14 18 22 26 30
𝑦 18 12 24 6 30 36
OR
1
Q.3 (a) Find coefficient of correlation between 𝑥 and 𝑦 if the regression lines are: 𝑥 + 03
6𝑦 = 6 and 3𝑥 + 2𝑦 = 10.
(b) Calculate the first four moments from the following data: 04
𝑥 0 1 2 3 4 5 6 7 8
𝑦 5 10 15 20 25 20 15 10 5
(c) The following data give the experience of machine operators and their 07
performance rating as given by the number of good parts turned out per 100 piece.
Operator 1 2 3 4 5 6
Performance 23 43 53 63 73 83
rating (𝑥)
Experience 5 6 7 8 9 10
(𝑦)
Calculate the regression line of performance rating on experience and also
estimate the probable performance if an operator has 11 years of experience.
Q.4 (a) Explain the term related to testing of hypothesis: (i) Null hypothesis (ii) Alternate 03
hypothesis and (iii) Level of Significance
(b) A dice is tossed 960 times and it falls with 5 upwards 184 times. Is the dice 04
unbiased at a level of significance of 0.01? (|𝑍0.01 | = 2.58)
(c) A set of five similar coins is tossed 320 times and result is obtained as follows 07
No of 0 1 2 3 4 5
heads
Frequency 6 27 72 112 71 32
Test the hypothesis that the data follow a binomial distribution. (Critical value
2
𝜒0.05 = 11.07)
OR
Q.4 (a) The heights of 10 males of a given locality are found to be 175, 168, 155, 170, 03
152, 170,175,160,160 and 165 cm. Based on this sample, find the 95% confidence
limits for the heights of males in that locality. [𝑡0.05 (𝑣 = 9) = 2.262]
(b) Random samples drawn from two countries gave the following data relating to the 04
heights of adult males:
Country A Country B
Standard 2.58 2.50
deviation (in
inches)
Number in 1000 1200
samples
Is the difference between the standard deviation significant? (|𝑍0.05 | = 1.96)
(c) If a random variable has a Poisson distribution such that 𝑃(𝑋 = 1) = 𝑃(𝑋 = 2), 07
find (i) the mean of the distribution (ii) 𝑃(𝑋 = 4) (iii) 𝑃(𝑋 ≥ 1) (iv)
𝑃(1 < 𝑋 < 4).
Q.5 (a) Fit a straight line𝑦 = 𝑎𝑥 + 𝑏 to the following data: 03
𝑥 1 2 3 4 6 8
𝑦 2.4 3 3.6 4 5 6
𝑥
(b) Fit a curve 𝑦 = 𝑎𝑏 to the following data: 04
𝑥 1 2 3 4 5 6 7 8
𝑦 1 1.2 1.8 2.5 3.6 4.7 6.6 9.1
(c) The lifetime of a certain kind of batteries has a mean life of 400 hours and the 07
standard deviation as 45 hours. Assuming the distribution of lifetime to be normal,
find (i) the percentage of batteries with a lifetime of at least 470 hours (ii) the
proportion of batteries with a lifetime between 385 and 415 hours, and (iii) the
minimum life of the best 5% of batteries. (𝑃(0 < 𝑧 < 0.33) = 0.1293)
OR
Q.5 (a) The mean and variance of a binomial variate are 8 and 6. Find 𝑃(𝑋 ≥ 2). 03
2
(b) Out of 800 families with 5 children each, how many would you except to have (i) 04
3 boys? (ii) 5 girls?
(c) Fit a second-degree parabolic curve to the following data: 07
𝑥 1 2 3 4 5 6 7 8 9
𝑦 2 6 7 8 10 11 11 10 9
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