Solution for Problem Set 3

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MAT204: Probability & Statistics
March 24, 2015

Problem 1: [3, Exercise 2.38]

A coin is tossed three times. If X is a random variable giving the number of heads that
arise, construct a table showing the probability distribution of X.
Solution:
x 0 1 2 3
f (x) 1/8 3/8 3/8 1/8

Problem 2: [3, Exercise 2.39]

An urn holds 5 white and 3 black marbles. If 2 marbles are to be drawn at random
without replacement and X denotes the number of white marbles, find the probability
distribution for X.
Solution:
x 0 1 2
f (x) 3/28 15/28 5/14

Problem 3: [3, Exercise 2.40]

Work Problem 2 if the marbles are to be drawn with replacement.
Solution:
x 0 1 2
f (x) 9/64 15/32 25/64

Problem 4: [3, Exercise 2.41]

Let Z be a random variable giving the number of heads minus the number of tails in 2
tosses of a fair coin. Find the probability distribution of Z. Compare with the results
of Examples 2.1 and 2.2.
Solution:
x -2 0 2
f (x) 1/4 1/2 1/4

1
Note:
x: random variables
f (x): probability functions

Problem 5: [3, Exercise 2.42]

Let X be a random variable giving the number of aces in a random draw of 4 cards from
an ordinary deck of 52 cards. Construct a table showing the probability distribution of
X.
Solution:
x 0 1 2 3 4
C40 C48
4
C41 C48
3
C42 C48
2
C43 C48
1
C44 C48
0
f (x) 4 4 4 4 4
C52 C52 C52 C52 C52

Problem 6: [3, Exercise 2.43]

The probability function of a random variable X is shown in the following table. Con-
struct a table giving the distribution function of X
x 1 2 3
f (x) 1/2 1/3 1/6

Solution:
x 1 2 3
F (x) 1/2 5/6 1

Problem 7: [3, Exercise 2.44]

Obtain the distribution for (a) Problem 1, (b) Problem 2, (c) Problem 3.
Solution:
a) 

 0 −∞ < x < 0

1/8 0≤x<1



F (x) = 1/2 1≤x<2

7/8 2≤x<3




1 3≤x<∞

b)

0
 −∞ < x < 0

3/28 0≤x<1
F (x) =


 18/28 1≤x<2
2≤x<∞

1

2
c)


 0 −∞ < x < 0

9/64 0≤x<1
F (x) =


 39/64 1≤x<2
2≤x<∞

1

Problem 8: [3, Exercise 2.45]

Obtain the distribution function for (a) Problem 4, (b) Problem 5.
Solution:
a)


 0 −∞ < x < −2

1/4 −2 ≤ x < 0
F (x) =


 3/4 0 ≤ x < 2
2≤x<∞

1

b) 

 0 −∞ < x < 0
 4 4
C48 /C52 0≤x<1




C 4 + C 1 C 3 /C 4

1≤x<2
48 4 48 52
F (x) =


 C48 + C4 C48 + C42 C48
4 1 3 2 4
/C52 2≤x<3
4 3
+ C41 C48 2
+ C42 C48 1
+ C43 C48 4
3≤x<4



 C48 /C52

1 4≤x<∞


Problem 9: [3, Exercise 2.46]

The following table shows the distribution function of a random variable X. Determine
(a) the probability function, (b) P(1 ≤ X ≤ 3), (c) P(X ≥ 2), (d) P(X < 3), (e)
P(X > 1.4).

x 1 2 3 4
F (x) 1/8 3/8 3/4 1

Solution:
a)

x 1 2 3 4
f (x) 1/8 1/4 3/8 1/4

b)
P (1 ≤ x ≤ 3) = 3/4

c)
P (x ≥ 2) = 7/8

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d)
P (x < 3) = 3/8

e)
P (x > 1.4) = 7/8

Problem 10: [3, Exercise 2.47]

A random variable X has density function:
 −3x
ce x>0
f (x) =
0 x≤0
Find (a) the constant c, (b) P (1 < X < 2), (c) P (X ≥ 3), (d) P (X < 1).
Solution:
a) +∞
Z ∞
−3x c −3x 
ce dx = − e
−∞ 3 
0
c c −3x c
= − lim e = =1⇒c=3
3 x→+∞ 3 3

b) 2
Z 2
−3x −3x  = e−3 − e−6

P (1 < X < 2) = 3e dx = e 
1 1

c) +∞
Z +∞
3e−3x dx = −e−3x  = e−9

P (X ≥ 3) = 
3 3

d)
Z +∞
P (X < 1) = 1 − P (X ≥ 1) = 1 − 3e−3x dx = 1 − lim e−3x + e−3 = 1 − e−3
1 x→+∞

Problem 11: [3, Exercise 2.48]

Find the distribution function for the random variable of Problem 10. Graph the den-
sity and distribution functions, describing the relationship between them.
Solution: ( (
3e−3x x > 0 1 − e−3x x > 0
f (x) = ⇒ F (x) =
0 x≤0 0 x≤0

Problem 12: [3, Exercise 2.49]

A random variable X has density function
 2
 cx 1 ≤ x ≤ 2
f (x) = cx 2 < x < 3
0 otherwise


4
Find (a) the constant c, (b) P (X > 2), (c) P (1/2 < X < 3/2).
Solution: 
 c0
 x≤1
 3
 3x 1 ≤ x ≤ 2
 2 
 cx 1 ≤ x ≤ 2


f (x) = cx 2 < x < 3 ⇒ F (x) =

0 otherwise
 c 2
x 2≤x≤3


 2



1 x≥3

a)
x < 1 ⇒ F (x) = 0
cx3 c
1 ≤ x ≤ 2 ⇒ F (x) = = (23 − 1)
3 3
2
cx c
2 < x < 3 ⇒ F (x) = = (32 − 22 )
2 2
Z ∞ Z 1 Z 2 Z 3 Z ∞
f (x)dx = f (x)dx + f (x)dx + f (x)dx + f (x)dx
−∞ −∞ 1 2 3
Z 2 Z 3
c c 29 6
= f (x)dx + f (x)dx = (23 − 1) + (32 − 22 ) = c = 1 ⇒ c =
1 2 3 2 6 29

b)
P (X > 2) = P (2 < X < 3) + P (3 < X)
1 6 2
∗ (3 − 22 ) + 0)
=
2 29
6 1 2 15
= 1 − P (X − 2) = 1 − ∗ (2 − 1) =
29 3 29

c)      
1 3 1 3
P <X< =P <X <1 +P 1<X <
2 2 2 2
 3 !
6 1 3 19
=0+ ∗ − 13 =
29 3 2 116

Problem 13: [3, Exercise 2.50]

Find the distribution function for the random variable X of Problem 12.
Solution: 

 0 x≤1
3
cx


+ k1 1 ≤ x ≤ 2



 3

F (x) =
cx2


+ k2 2 ≤ x ≤ 3



 2



1 x≥3

5
 c c 6/29 6/29
(1) : x3 + k1 = x2 + k2 ⇒ ∗ 23 + k1 = ∗ 22 + k2
3 2 3 2
16 12 4
⇒ + k1 = + k2 ⇒ k2 = k1 +
29 29 29
c c 6/29 6/29
(2) : x3 + k1 + x2 + k2 = 1 ⇒ ∗ 13 + k1 + ∗ 32 + k2 = 1
3 2 3 2
2 27
⇒ + k1 + + k2 = 1 ⇒ 1 + k1 + k2 = 1 ⇒ k1 + k2 = 0 ⇒ k1 = −k2
29 29
From (1)
 and (2):
4 2 2
k1 = − k1 + ⇒ k1 = − ; k2 =
29 29 29
Therefore: 

 0 x≤1
2x3 − 2


1≤x≤2



 29

F (x) =
 3x2 + 2



 2≤x≤3
 29



1 x≥3

Problem 14: [3, Exercise 2.51]

The distribution function of a random variable X is given by
 3
 cx 0 ≤ x < 3
F (x) = 1 x≥3
0 x<0


If P (X = 3) = 0, find (a) the constant c, (b) the density function, (c) P (X > 1), (d)
P (1 < X < 2).
Solution:  3
 cx 0 ≤ x < 3 
3cx2 0 ≤ x < 3
F (x) = 1 x≥3 ⇒ f (x) =
0 otherwise
0 x<0


a) 3
Z ∞ Z 3
3
 3 1
f (x)dx = 1 ⇒ cx dx = 1 ⇒ cx  = 1 ⇒ c =
−∞ 0 0 27

b) 
 1 x2 0≤x≤3
f (x) = 9
0 otherwise

c) Z 3
1 3 26
P (X > 1) = f (x)dx = (3 − 1) =
1 27 27

d) Z 2
1 3 7
P (1 < X < 2) = f (x)dx = (2 − 1) =
1 27 27

6
Problem 15: [3, Exercise 2.52]
Can the function 
c(1 − x2 ) 0 ≤ x ≤ 1
F (x) =
0 otherwise
be a distribution function? Explain.
Solution:
( (
c(1 − x2 ) 0 ≤ x ≤ 1 −2cx 0 ≤ x ≤ 1
F (x) = ⇒ f (x) =
0 otherwise 0 otherwise
Z ∞
1

2
f (x)dx = 1 ⇒ c(1 − x )  = 1
−∞ 0
2
⇔ c − c(1 − 0) = 1 ⇔ c − c = 1 ⇔ 0 = 1

Therefore, the given function can’t be a distribution function.

Problem 16: [3, Exercise 2.53]

Let X be a random variable having density function

cx 0 ≤ x ≤ 2
f (x) =
0 otherwise

1 3
Find (a) the value of the constant c, (b) P ( < X < ), (c) P (X > 1), (d) the
2 2
distribution function.
Solution:
a) 2
Z ∞ Z 2
cx2 
f (x)dx = 1 ⇒ cxdx = 1 ⇒ =1
−∞ 0 2 0
 2 
2 1
⇔c − 0 = 1 ⇔ 2c = 1 ⇔ c =
2 2

b)
3/2 3/2  2  2 !
cx2  x2 
 
1 3 1 3 1 1
P <X< = = = − =
2 2 2 1/2 4 1/2 4
  2 2 2

c) 2
x2  1 2
3
P (X > 1) = = 2 − 12 =
4 1 4
 4

d) 

 0 x<0
 2
x
F (x) = 0≤x≤2
4


1 x>2

7
Problem 17: [2, Exercise 3, section 3.1]
Suppose that two balanced dice are rolled, and let X denote the absolute value of the
difference between the two numbers that appear. Determine and sketch the p.f. of X.
Solution:
There are total 36 possible outcomes when 2 dice are rolled because each die has 6 sides
(from 1 to 6).
• X = 0 for 6 outcomes

• X = 1 for 10 outcomes

• X = 2 for 8 outcomes

• X = 3 for 6 outcomes

• X = 4 for 4 outcomes

• X = 5 for 2 outcomes

Therefore, the p.f. f (x) is as follows:

x 0 1 2 3 4 5
f (x) 3/18 5/18 4/18 3/18 2/18 1/18

Problem 18: [2, Exercise 4, section 3.1]

Suppose that a fair coin is tossed 10 times independently. Determine the p.f. of the
number of heads that will be obtained.
Solution:
For x = 0,1,...,10, the probability of obtaining exactly x heads is:
   10
10 1
x 2

Problem 19: [2, Exercise 5, section 3.1]

Suppose that a box contains seven red balls and three blue balls. If five balls are se-
lected at random, without replacement, determine the p.f. of the number of red balls
that will be obtained.
Solution:
For x = 2,3,4,5, the probability that obtaining x red balls is:
    
7 3 10
/
x 5−x 5

Problem 20: [2, Exercise 6, section 3.1]

Suppose that a random variable X has the binomial distribution with parameters n =

8
15 and p = 0.5. Find P(X < 6).
Solution:
The desired probability is the sum of the entries for k = 0,1,2,3,4 and 5 in that part of
the table of binomial probabilities corresponding to n = 15 and p = 0.5. The sum is:

0.1509

Problem 21: [2, Exercise 4, section 3.2]

Suppose that the p.d.f. of a random variable X is as follows:
 2
cx 1 ≤ x ≤ 2
f (x) =
0 otherwise.

a. Find the value of the constant c and sketch the p.d.f.

Solution:
We have: Z ∞ Z 2 2
2 cx3  7
f (x)dx = cx dx = = c=1
−∞ 1 3 1 3

3
⇒c=
7

Therefore, the p.d.f is as follows:

b. Find the value of P(X > 3/2).

Solution: Z 2 2 2
2
cx3  x3 
Z
2 37
f (x)dx = cx dx = = =
3/2 3/2 3 3/2 7 3/2 56
 

Problem 22: [2, Exercise 5, section 3.2]

Suppose that the p.d.f. of a random variable X is as follows:

1/8x 0 ≤ x ≤ 4
f (x) =
0 otherwise.

9
a. Find the value of t such that P(X ≤ t) = 1/4.
Solution: Z t t
1 1 2  1 t2 1
xdx = x  = ⇒ = ⇒t=2
0 8 16 0 4 16 4
b. Find the value of t such that P(X ≥ t) = 1/2.
Solution: 4
Z 4
1 1 2  1 t2 1 √
xdx = x  = ⇒ 1 − = ⇒t= 8
t 8 16 t 2 16 2

Problem 23: [1, Exercise 3.5]

Determine the value c so that each of the following functions can serve as a probability
distribution of the discrete random variable X :
a. f (x) = c(x2 + 4), for x = 0, 1, 2, 3;
Solution:
X 3
c(x2 + 4) = 30c = 1
x=0

1
⇒c=
30
  
2 3
b. f (x) = c , for x = 0, 1, 2.
x 3−x
Solution:
2            
X 2 3 2 3 2 3 2 3
c =c + + = 10c = 1
x=0
x 3−x 0 3 1 2 2 1

1
⇒c=
10

Problem 24: [1, Exercise 3.30]

Measurements of scientific systems are always subject to variation, some more than
others. There are many structures for measurement error, and statisticians spend a
great deal of time modeling these errors. Suppose the measurement error X of a certain
physical quantity is decided by the density function

k(3 − x2 ) −1 ≤ x ≤ 1,
f (x) =
0, elsewhere.

a) Determine k that renders f(x) a valid density function.

Solution: Z 1  1
x3 

2 16
1=k (3 − x )dx = k 3x − = k
−1 3 
−1 3
3
⇒k=
16
b) Find the probability that a random error in measurement is less than 1/2.
Solution:

10
For −1 ≤ x ≤ 1 :
x  x
x3
Z 
3 2 1 3  1 9
F (x) = (3 − t )d(t) = 3t − t  = + x −
16 −1 3 −1 2 16 16
    3
1 1 9 1 1 1 99
⇒ P (X < ) = − − =
2 2 16 2 16 2 128
c) For this particular measurement, it is undesirable if the magnitude of the error (i.e.,
|x|) exceeds 0.8. What is the probability that this occurs?
Solution:

P (|X| < 0.8) = P (X < −0.8) + P (X > 0.8) = F (−0.8) + 1 − F (0.8)

   
1 9 1 3 1 9 1 3
=1+ − 0.8 + 0.8 − + 0.8 − 0.8 = 0.164
2 16 16 2 16 16

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References
[1] Walpole, R. E., Myers, R. H., Myers, S. L. and Ye, K., Probability &
Statistics for Engineers & Scientists, 9th ed., MA, USA: Prentice-Hall, 2012.

[2] DeGroot, M. H. and Schervish, M. J., Probability and Statistics, 4th ed., MA,
USA: Pearson Education, Inc., 2012.

[3] Murray, R. S., John, J. S. and R, A. Srinivasan, Probability and Statistics,

3rd ed., USA: McGraw-Hill, 2009.

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