Homework Problem 1.7
A randomly selected car battery is tested and the time of failure is recorded. Give an
appropriate sample space for this experiment.
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical
Statistics.)
Solutions:

Homework Problem 1.27
A bag contains five blue balls and three red balls. A boy draws a ball, and then draws
another without replacement. Compute the following probabilities:
(a) P ( 2 blue balls ) .
(b) P (1 blue and 1 red ) .
(c) P ( at least 1 blue ) .

Homework Problem 1.37
Let P ( A ) = 0.4 and P ( A B ) = 0.6 .
(a) For what value of P ( B ) are A and B mutually exclusive?
(b) For what value of P ( B ) are A and B independent?
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematica

Homework Problem 1.19
Let P ( A) = P ( B ) = 1/ 3 and P ( A B ) = 1/10 . Find the following:
(a) P ( B ) .
(b) P ( A B ) .
(c) P ( B A ) .
(d) P ( A B ) .
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical
Statistics.)
Solutions

Homework Problem 1.15
Two part-time teachers are hired by the mathematics department and each is assigned at
random to teach a single course, in trigonometry, algebra, or calculus. List the outcomes
in the sample space and find the probability that they w

Homework Problem 1.33
One card is selected from a deck of 52 cards and placed in a second deck. A card then is
selected from the second deck.
(a) What is the probability the second card is an ace?
(b) If the first card is placed into a deck of 54 cards co

Homework Problem 1.35
In a bolt Factory, machines 1, 2, and 3 respectively produce 20%, 30%, and 50% of the
total output. Of their respective outputs, 5%, 3%, and 2% are defective. A bolt is
selected at random.
(a) What is the probability that it is defec

Homework Problem 4.3
Five cards are drawn without replacement from a regular deck of 52 cards. Let X
represent the number of aces, Y the number of kings, and Z the number of queens
obtained. Give the probability of each of the following events:
(a) A = [

Homework Problem 5.1
Let X 1 , X 2 , X 3 , and X 4 be independent random variables, each having the same
distribution with mean 5 and standard deviation 3, and let
Y = X1 + 2 X 2 + X 3 X 4 .
(a) Find E (Y ) .
(b) Find Var (Y ) .
(Lee J. Bain and Max Engel

Homework Problem 6.1
Let X be a random variable with pdf
4 x3 , 0 < x < 1
fX ( x) =
.
otherwise
0,
Use the cumulative (CDF) technique to determine the pdf of each of the following
random variables:
(a) Y = X 4 .
(b) W = e X .
(c) Z = ln X .
(d) U = ( X 0

Homework Problem 6.7
Let U ~ UNIF ( 0,1) . Find transformations y = G1 ( u ) and w = G2 ( u ) such that
(a) Y = G1 (U ) ~ EXP (1) .
(b) W = G2 (U ) ~ BIN ( 3,1/ 2 ) .
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical
Statistics

Homework Problem 4.5
Rework Exercise 3, assuming that the cards were drawn with replacement.
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical
Statistics.)
Exercise 3 is repeated here for convenience.
Five cards are drawn witho

Homework Problem 4.7
Suppose that X 1 and X 2 are discrete random variables with joint pdf of the form
c ( x1 + x2 ) , x1 = 0,1, 2, x2 = 0,1, 2
f X1 , X 2 ( x1 , x2 ) =
.
otherwise
0,
Find the constant c.
(Lee J. Bain and Max Engelhardt, Introduction to

Homework Problem 2.5
A discrete random variable has pdf f X ( x ) .
1 x
, x = 1, 2,3
k
, find k .
(a) If f X ( x ) = 2
0,
otherwise
1 x 1
, x = 0,1, 2
k
(b) Is a function of the form f X ( x ) = 2 2
a pdf for any k ?
otherwise
0,
(Lee J. Bain and M

Homework Problem 1.9
In Exercise 2, suppose that each of the nine possible outcome sin the sample space is
equally likely to occur. Compute each of the following:
(a) P(both red).
(b) P ( C1 ) .
(c) P ( C2 ) .
(d) P ( C1 C2 ) .
(e) P ( C1 C2 ) .
(f) P ( C

Homework Problem 6.3
The measured radius of a circle, R , has pdf
6r (1 r ) , 0 < r < 1
fR ( r ) =
.
otherwise
0,
(a) Find the distribution of the circumference.
(b) Find the distribution of the area of the circle.
(Lee J. Bain and Max Engelhardt, Introd

Homework Problem 3.5
(a) The Chevalier de Mere used to bet that he would get at least one 6 in four rolls of a
die. Was this a good bet?
(b) He also bet that he would get at least one pair of 6s in 24 rolls of two dice.
What was his probability of winning

Homework Problem 5.7
Suppose X and Y are independent random variables with E X ( X ) = 2, EY (Y ) = 3,
VarX ( X ) = 4, and VarY (Y ) = 16 .
(a) Find E ( 5 X Y ) .
(b) Find Var ( 5 X Y ) .
(c) Find Cov ( 3 X + Y , Y ) .
(d) Find Cov ( X ,5 X Y ) .
(Lee J.

Homework Problem 3.9
An office has 10 employees, three mean and seven women. The manager chooses four
at random to attend a short course on quality improvement.
(a) What is the probability that an equal number of men and women are chosen?
(b) What is the

Homework Problem 1.13
When an experiment is performed, one and only one of the events A 1 , A 2 , or A 3 will
occur. Find P ( A 1 ) , P ( A 2 ) , and P ( A 3 ) under each of the following assumptions:
(a) P ( A 1 ) = P ( A 2 ) = P ( A 3 ) .
(b) P ( A 1 )

Homework Problem 1.1
A gum-ball machine gives out a red, a black, or a green gum ball.
(a) Describe an appropriate sample space.
(b) List all possible events.
(c) If R is the event red, then list the outcomes in R .
(d) If G is the event green, then what

Homework Problem 4.9
Let X 1 and X 2 be discrete random variables with joint pdf f X1 , X 2 ( x1 , x2 ) given by the
following table:
1
2
3
X1
X2
2
1/6
1/9
1/4
1
1/12
0
1/18
3
0
1/5
2/15
(a) Find the marginal pdfs of X 1 and X 2 .
(b) Are X 1 and X 2 inde

Homework Problem 5.3
Suppose X and Y are continuous random variables with joint pdf
24 xy, 0 < x, 0 < y, x + y < 1
f X ,Y ( x, y ) =
.
otherwise
0,
(a) Find E ( XY ) .
(b) Find the covariance of X and Y .
(c) Find the correlation coefficient of X and Y .

Homework Problem 1.3
There are four basic blood groups: O, A, B, and AB. Ordinarily, anyone can receive the
blood of a donor from their own group. Also, anyone can receive the blood of a donor
from the O group, and any of the four types can be used by a r

Homework Problem 2.1
Let e = ( i, j ) represent an arbitrary outcome resulting from two rolls of the four-sided
die of Example 2.1.1. Tabulate the discrete pdf and sketch the graph of the CDF
for the following random variables:
(a) Y ( e ) = i + j .
(b) Z

Homework Problem 2.7
c ( 8 x ) , x = 0,1, 2, 3, 4,5
A discrete random variable X has a pdf of the form f X ( x ) =
.
otherwise
0,
(a) Find the constant c.
(b) Find the CDF, FX ( x ) .
(c) Find P [ X > 2] .
(d) Find E ( X ) .
(Lee J. Bain and Max Engelhar

Homework Problem 6.5
Prove Theorem 6.3.4, assuming that the CDF FX ( x ) is a one-to-one function.
(Lee J. Bain and Max Engelhardt, Introduction to Probability and Mathematical
Statistics.)
Solution:
Recall that Theorem 6.3.4 states: Let FX ( x ) be a CDF