178
Self-Test Exercises for Chapter 6
Law of total probability
Case I: Discrete
pX (x) =
X
pX|Y (x | y)pY (y)
y
Case II: Continuous
fX (x) =
Z +∞
−∞
fX|Y (x | y)fY (y) dy
Self-Test Exercises for Chapter 6
For each of the following multiple-choice questions, choose the best response
among those provided. Answers can be found in Appendix B.
S6.1 Let X1 , X2 , X3 , X4 be independent and identically distributed random variables, each with P (Xi = 1) = 12 and P (Xi = 0) = 12 . Let P (X1 + X2 +
X3 + X4 = 3) = r, then the value of r is
(A)
1
16
(B)
1
4
(C)
1
2
(D) 1
(E) none of the above.
S6.2 Let (X, Y ) be a discrete random vector with joint probability mass function
given by
pX,Y (0, 0) = 1/4
pX,Y (0, 1) = 1/4
pX,Y (1, 1) = 1/2
Then P (Y = 1) equals
(A) 1/3
(B) 1/4
(C) 1/2
Self-Test Exercises for Chapter 6
179
(D) 3/4
(E) none of the above.
S6.3 Let (X, Y ) be a random vector with joint probability mass function given by
pX,Y (1, −1) =
1
4
pX,Y (1, 1) =
1
2
pX,Y (0, 0) =
1
4
Let pX (x) denote the marginal probability mass function for X . The value
of pX (1) is
(A) 0
(B)
1
4
(C)
1
2
(D) 1
(E) none of the above.
S6.4 Let (X, Y ) be a random vector with joint probability mass function given by
pX,Y (1, −1) =
1
4
pX,Y (1, 1) =
1
2
pX,Y (0, 0) =
1
4
The value of P (XY > 0) is
(A) 0
(B)
1
4
(C)
1
2
(D) 1
(E) none of the above.
S6.5 Let (X, Y ) be a continuous random vector with joint probability density
function
(
fX,Y (x, y) =
Then P (X > 0) equals
(A) 0
(B) 1/4
1/2 if −1 ≤ x ≤ 1 and 0 ≤ y ≤ 1
0
otherwise
180
Self-Test Exercises for Chapter 6
(C) 1/2
(D) 1
(E) none of the above.
S6.6 Suppose (X, Y ) is a continuous random vector with joint probability density
function
(
e−y if 0 ≤ x ≤ 1 and y ≥ 0
fX,Y (x, y) =
0
otherwise
Then P (X > 12 ) equals
(A) 0
(B) 1/2
(C) e−1/2
(D)
1 −y
2e
(E) none of the above.
S6.7 Let (X, Y ) be a discrete random vector with joint probability mass function
given by
pX,Y (0, 0) = 1/3
pX,Y (0, 1) = 1/3
pX,Y (1, 0) = 1/3
Then P (X + Y = 1) equals
(A) 0
(B) 1/3
(C) 2/3
(D) 1
(E) none of the above.
S6.8 Let (X, Y ) be a discrete random vector with joint probability mass function
given by
pX,Y (0, 0) = 1/3
pX,Y (0, 1) = 1/3
pX,Y (1, 0) = 1/3
Self-Test Exercises for Chapter 6
181
Let W = max{X, Y }. Then P (W = 1) equals
(A) 0
(B) 1/3
(C) 2/3
(D) 1
(E) none of the above.
S6.9 Suppose (X, Y ) is a continuous random vector with joint probability density
function


if 0 ≤ x ≤ 2 and
 y
0≤y≤1
fX,Y (x, y) =

 0
otherwise
Then E(X 2 ) equals
(A) 0
(B) 1
(C) 4/3
(D) 8/3
(E) none of the above.
S6.10 Suppose (X, Y ) is a continuous random vector with joint probability density
function


if 0 ≤ x ≤ 2 and
 y
0≤y≤1
fX,Y (x, y) =

 0
otherwise
Then P (X < 1, Y < 0.5) equals
(A) 0
(B) 1/8
(C) 1/4
(D) 1/2
(E) none of the above.
182
Self-Test Exercises for Chapter 6
S6.11 The weights of individual oranges are independent random variables, each
having an expected value of 6 ounces and standard deviation of 2 ounces.
Let Y denote the total net weight (in ounces) of a basket of n oranges. The
variance of Y is equal to
√
(A) 4 n
(B) 4n
(C) 4n2
(D) 4
(E) none of the above.
S6.12 Let X1 , X2 , X3 be independent and identically distributed random variables,
each with P (Xi = 1) = p and P (Xi = 0) = 1 − p. If P (X1 + X2 + X3 =
3) = r, then the value of p is
(A)
1
3
(B)
1
2
(C)
√
3
r
(D) r3
(E) none of the above.
S6.13 If X and Y are random variables with Var(X) = Var(Y ), then Var(X + Y )
must equal
(A) 2Var(X)
√
(B) 2(Var(X))
(C) Var(2X)
(D) 4Var(X)
(E) none of the above.
S6.14 Let (X, Y ) be a continuous random vector with joint probability density
function
(
1/π if x2 + y 2 ≤ 1
fX,Y (x, y) =
0
otherwise
Then P (X > 0) equals
Self-Test Exercises for Chapter 6
183
(A) 1/(4π)
(B) 1/(2π)
(C) 1/2
(D) 1
(E) none of the above.
S6.15 Let X be a random variable. Then E[(X + 1)2 ] − (E[X + 1])2 equals
(A) Var(X)
(B) 2E[(X + 1)2 ]
(C) 0
(D) 1
(E) none of the above.
S6.16 Let X , Y and Z be independent random variables with
E(X) = 1
E(Y ) = 0
E(Z) = 1
Let
W = X(Y + Z).
Then E(W ) equals
(A) 0
(B) E(X 2 )
(C) 1
(D) 2
(E) none of the above
S6.17 Toss a fair die twice. Let the random variable X represent the outcome of
the first toss and the random variable Y represent the outcome of the second
toss. What is the probability that X is odd and Y is even.
(A) 1/4
(B) 1/3
(C) 1/2
184
Self-Test Exercises for Chapter 6
(D) 1
(E) none of the above
S6.18 Assume that (X, Y ) is random vector with joint probability density function
(
fX,Y (x, y) =
k for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and y ≤ x
0 otherwise
where k is a some positive constant. Let W = X −Y . To find the cumulative
distribution function for W we can compute
FW (w) =
Z BZ D
A
C
k dx dy
for 0 ≤ w ≤ 1. The limit of integration that should appear in position D is
(A) min{1, x − w}
(B) min{1, y − w}
(C) min{1, x − y}
(D) min{1, y + w}
(E) none of the above.
S6.19 Assume that (X, Y ) is a random vector with joint probability mass function
given by
pX,Y (−1, 0) = 1/4
pX,Y (0, 0) = 1/2
pX,Y (0, 1) = 1/4
Define the random variable W = XY . The value of P (W = 0) is
(A) 0
(B) 1/4
(C) 1/2
(D) 3/4
(E) none of the above.
Self-Test Exercises for Chapter 6
185
S6.20 Let X be a random variable with E(X) = 0 and Var(X) = 1. Let Y be a
random variable with E(Y ) = 0 and Var(Y ) = 4. Then E(X 2 + Y 2 ) equals
(A) 0
(B) 1
(C) 3
(D) 5
(E) none of the above.
S6.21 Suppose (X, Y ) is a continuous random vector with joint probability density
function
(
fX,Y (x, y) =
·
1
Then E
XY
4xy
0
if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
otherwise
¸
equals
(A) +∞
(B) 0
(C) 1/2
(D) 4
(E) none of the above.
S6.22 The life lengths of two transistors in an electronic circuit is a random vector
(X, Y ) where X is the life length of transistor 1 and Y is the life length of
transistor 2. The joint probability density function of (X, Y ) is given by
(
fX,Y (x, y) =
2e−(x+2y) if x ≥ 0 and y ≥ 0
0
otherwise
Then P (X + Y ≤ 1) equals
(A)
(B)
(C)
R1R1
0
0
2e−(x+2y) dx dy
R 1 R 1−y
0
0
0
y
R1R1
2e−(x+2y) dx dy
2e−(x+2y) dx dy
186
Questions for Chapter 6
(D)
R1R1
0
1−y
2e−(x+2y) dx dy
(E) none of the above.
Questions for Chapter 6
6.1 A factory can produce two types of gizmos, Type 1 and Type 2. Let X be a
random variable denoting the number of Type 1 gizmos produced on a given
day, and let Y be the number of Type 2 gizmos produced on the same day.
The joint probability mass function for X and Y is given by
pX,Y (1, 0) = 0.10;
pX,Y (1, 2) = 0.10;
pX,Y (2, 1) = 0.20;
pX,Y (3, 1) = 0.05;
pX,Y (1, 1) = 0.20;
pX,Y (1, 3) = 0.10;
pX,Y (2, 2) = 0.20;
pX,Y (3, 2) = 0.05;
(a) Compute P (X ≤ 2, Y = 2), P (X ≤ 2, Y 6= 2) and
P (Y > 0).
(b) Find the marginal probability mass functions for X and for
Y.
(c) Find the distribution for the random variable Z = X + Y
which is the total daily production of gizmos.
6.2 A sample space Ω is a set consisting for four points, {ω1 , ω2 , ω3 , ω4 }. A
probability measure, P (·) assigns probabilities, as follows:
P ({ω1 }) =
P ({ω3 }) =
1
2
1
4
P ({ω2 }) =
P ({ω4 }) =
1
8
1
8
Random variables X , Y and Z are defined as
X(ω1 ) = 0,
Y (ω1 ) = 0,
Z(ω1 ) = 1,
X(ω2 ) = 0,
Y (ω2 ) = 1,
Z(ω2 ) = 2,
X(ω3 ) = 1,
Y (ω3 ) = 1,
Z(ω3 ) = 3,
X(ω4 ) = 1
Y (ω4 ) = 0
Z(ω4 ) = 4
(a) Find the joint probability mass function for (X, Y, Z).
(b) Find the joint probability mass function for (X, Y ).
(c) Find the probability mass function for X .
Questions for Chapter 6
187
(d) Find the probability mass function for the random variable
W = X + Y + Z.
(e) Find the probability mass function for the random variable
T = XY Z .
(f) Find the joint probability mass function for the random variable (W, T ).
(g) Find the probability mass function for the random variable
V = max{X, Y }.
6.3 Let (X, Y ) have a uniform distribution over the rectangle
¡
¢
(0, 0), (0, 1), (1, 1), (1, 0) .
In other words, the probability density function for (X, Y ) is given by
(
1
0
fX,Y (x, y) =
if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
otherwise
Find the cumulative distribution function for the random variable Z = X+Y .
6.4 Two light bulbs are burned starting at time 0. The first one to fail burns out
at time X and the second one at time Y . Obviously X ≤ Y . The joint
probability density function for (X, Y ) is given by
(
fX,Y (x, y) =
2e−(x+y)
0
if 0 < x < y < ∞
otherwise
(a) Sketch the region in (x, y)-space for which the above probability density function assigns positive mass.
(b) Find the marginal probability density functions for X and
for Y .
(c) Let Z denote the excess life of the second bulb, i.e., let
Z = Y − X . Find the probability density function for Z .
(d) Compute P (Z > 1) and P (Y > 1).
6.5 If two random variables X and Y have a joint probability density function
given by
(
fX,Y (x, y) =
2
0
if x > 0, y > 0 and x + y < 1
otherwise
188
Questions for Chapter 6
(a) Find the probability that both random variables will take on
a value less than 12 .
(b) Find the probability that X will take on a value less than
and Y will take on a value greater than 12 .
1
4
(c) Find the probability that the sum of the values taken on by
the two random variables will exceed 23 .
(d) Find the marginal probability density functions for X and
for Y .
(e) Find the marginal probability density function for Z = X +
Y.
6.6 Suppose (X, Y, Z) is a random vector with joint probability density function
(
fX,Y,Z (x, y, z) =
α(xyz 2 )
0
if 0 < x < 1, 0 < y < 2 and 0 < z < 2
otherwise
(a) Find the value of the constant α.
(b) Find the probability that X will take on a value less than
and Y and Z will both take on values less than 1.
1
2
(c) Find the probability density function for the random vector
(X, Y ).
6.7 Suppose that the random vector (X, Y ) has joint probability density function
(
fX,Y (x, y) =
kx(x − y)
0
if 0 ≤ x ≤ 2 and |y| ≤ x
otherwise
(a) Sketch the support of the distribution for (X, Y ) in the xy plane.
(b) Evaluate the constant k .
(c) Find the marginal probability density function for Y .
(d) Find the marginal probability density function for X .
(e) Find the probability density function for Z = X + Y .
Questions for Chapter 6
189
6.8 The length, X , and the width, Y , of salt crystals form a random variable
(X, Y ) with joint probability density function
(
fX,Y (x, y) =
x
0
if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2
otherwise
(a) Find the marginal probability density functions for X and
for Y .
6.9 If the joint probability density function of the price, P , of a commodity (in
dollars) and total sales, S , (in 10, 000 units) is given by
(
fS,P (s, p) =
5pe−ps
0
0.20 < p < 0.40 and s > 0
otherwise
(a) Find the marginal probability density functions for P and
for S .
(b) Find the conditional probability density function for S given
that P takes on the value p.
(c) Find the probability that sales will exceed 20, 000 units given
P = 0.25.
6.10 If X is the proportion of persons who will respond to one kind of mail-order
solicitation, Y is proportion of persons who will respond to a second type of
mail-order solicitation, and the joint probability density function of X and
Y is given by
(
fX,Y (x, y) =
2
5 (x
0
+ 4y)
0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
otherwise
(a) Find the marginal probability density functions for X and
for Y .
(b) Find the conditional probability density function for X given
that Y takes on the value y .
(c) Find the conditional probability density function for Y given
that X takes on the value x.
(d) Find the probability that there will be at least a 20% response
to the second type of mail-order solicitation.
190
Questions for Chapter 6
(e) Find the probability that there will be at least a 20% response
to the second type of mail-order solicitation given that there
has only been a 10% response to the first kind of mail-order
solicitation.
6.11 An electronic component has two fuses. If an overload occurs, the time when
fuse 1 blows is a random variable, X , and the time when fuse 2 blows is a
random variable, Y . The joint probability density function of the random
vector (X, Y ) is given by
(
fX,Y (x, y) =
e−y
0
0 ≤ x ≤ 1 and y ≥ 0
otherwise
(a) Compute P (X + Y ≥ 1).
(b) Are X and Y independent? Justify your answer.
6.12 The coordinates of a laser on a circular target are given by the random vector
(X, Y ) with the following probability density function:

 1
fX,Y (x, y) =
 π
0
x2 + y 2 ≤ 1
otherwise
Hence, (X, Y ) has a uniform distribution on a disk of radius one centered at
(0, 0).
(a) Compute P (X 2 + Y 2 ≤ 0.25).
(b) Find the marginal distributions for X and Y .
(c) Compute P (X ≥ 0, Y > 0).
(d) Compute P (X ≥ 0 | Y > 0).
(e) Find the conditional distribution for X given Y = y . (Note:
Be careful of the case |y| = 1.)
6.13 Let X and Y be discrete random variables with the support of X denoted by
Θ and the support of Y denoted by Φ. Let pX be the marginal probability
mass function for X , let pX|Y be the conditional probability mass function
of X given Y , and let pY |X be the conditional probability mass function of
Questions for Chapter 6
191
Y given X . Show that
pY |X (y|x)pX (x)
pX|Y (x|y) = X
pY |X (y|x)pX (x)
x∈Θ
for any x ∈ Θ and y ∈ Φ.
6.14 The volumetric fractions of each of two compounds in a mixture are random variables X and Y , respectively. The random vector (X, Y ) has joint
probability density function
(
2
0
fX,Y (x, y) =
0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 − x
otherwise
(a) Determine the marginal probability density functions for X
and for Y .
(b) Are X and Y independent random variables? Justify your
answer.
(c) Compute P (X < 0.5 | Y = 0.5).
(d) Compute P (X < 0.5 | Y < 0.5).
6.15 Suppose (X, Y ) is a continuous random vector with joint probability density
function
(
2
0
fX,Y (x, y) =
if x > 0, y > 0 and x + y < 1
otherwise
Find E(X + Y ).
6.16 Suppose (X, Y, Z) is a random vector with joint probability density function
(
fX,Y,Z (x, y, z) =
3
2
8 (xyz )
0
if 0 < x < 1, 0 < y < 2, 0 < z < 2
otherwise
Find E(X + Y + Z).
6.17 Suppose the joint probability density function of the price, P , of a commodity
(in dollars) and total sales, S , (in 10, 000 units) is given by
(
fS,P (s, p) =
Find E(S | P = 0.25).
5pe−ps
0
0.20 < p < 0.40 and s > 0
otherwise
192
Questions for Chapter 6
6.18 Let X be the proportion of persons who will respond to one kind of mailorder solicitation, and let Y be the proportion of persons who will respond
to a second type of mail-order solicitation. Suppose the joint probability
density function of X and Y is given by
(
fX,Y (x, y) =
2
5 (x
+ 4y)
0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
otherwise
0
Find E(X | Y = y) for each 0 < y < 1.
6.19 Suppose that (X, Y ) is a random vector with the following joint probability
density function:
(
fX,Y (x, y) =
2x
0
if 0 < x < 1 and 0 < y < 1
otherwise
Find the marginal probability density function for Y .
6.20 Suppose that (X, Y ) is a random vector with the following joint probability
density function:
(
fX,Y (x, y) =
2xe−y
0
if 0 ≤ x ≤ 1 and y ≥ 0
otherwise
(a) Find P (X > 0.5).
(b) Find the marginal probability density function for Y .
(c) Find E(X).
(d) Are X and Y independent random variables? Justify your
answer.
6.21 The Department of Metaphysics of Podunk University has 5 students, with
2 juniors and 3 seniors. Plans are being made for a party. Let X denote the
number of juniors who will attend the party, and let Y denote the number of
seniors who will attend.
After an extensive survey was conducted, it has been determined that the
random vector (X, Y ) has the following joint probability mass function:
pX,Y (x, y)
X (juniors)
0
1
2
Y (seniors)
0
1
2
.01 .25 .03
.04 .05 .15
.04 .12 .20
3
.01
.05
.05
Questions for Chapter 6
193
(a) What is the probability that 3 or more students (juniors
and/or seniors) will show up at the party?
(b) What is the probability that more seniors than juniors will
show up at the party?
(c) If juniors are charged $5 for attending the party and seniors
are charged $10, what is the expected amount of money that
will be collected from all students?
(d) The two juniors arrive early at the party. What is the probability that no seniors will show up?
6.22 Let (X, Y ) be a continuous random vector with joint probability density
function given by
(
fX,Y (x, y) =
2
0
if x ≥ 0, y ≥ 0 and x + y ≤ 1
.
otherwise
(a) Compute P (X < Y ).
(b) Find the marginal probability density function for X and the
marginal probability density function for Y .
(c) Find the conditional probability density function for Y given
that X = 12 .
(d) Are X and Y independent random variables? Justify your
answer.
6.23 Let (X, Y ) be a continuous random vector with joint probability density
function given by
(
fX,Y (x, y) =
4xy
0
for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
.
otherwise
(a) Find the marginal probability density function for X and the
marginal probability density function for Y .
(b) Are X and Y independent random variables? Justify your
answer.
(c) Find P (X + Y ≤ 1).
(d) Find P (X > 0.5 | Y ≤ 0.5).
194
Questions for Chapter 6
6.24 Let (X, Y ) be a continuous random vector with joint probability density
function given by
(
fX,Y (x, y) =
6x2 y
0
for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
otherwise
(a) Find the marginal probability density function for X and the
marginal probability density function for Y .
(b) Are X and Y independent random variables? Justify your
answer.
(c) Find P (X + Y ≤ 1).
(d) Find E(X + Y ).
6.25 A box contains four balls numbered 1, 2, 3 and 4. A game consists of drawing
one of the balls at random from the box. It is not replaced. A second ball is
then drawn at random from the box.
Let X be the number on the first ball drawn, and let Y be the number on the
second ball.
(a) Find the joint probability mass function for the random vector (X, Y ).
(b) Compute P (Y ≥ 3 | X = 1).
6.26 Suppose that (X, Y ) is a random vector with the following joint probability
density function:


2xe−y



fX,Y (x, y) =
if 0 < x < 1 and y > 0

0



otherwise
(a) Compute P (X > 0.5, Y < 1.0).
(b) Compute P (X > 0.5 | Y < 1.0).
(c) Find the marginal probability density function for X .
(d) Find E(X + Y ).
Questions for Chapter 6
195
6.27 Past studies have shown that juniors taking a particular probability course
receive grades according to a normal distribution with mean 65 and variance
36. Seniors taking the same course receive grades normally distributed with
a mean of 70 and a variance of 36. A probability class is composed of 75
juniors and 25 seniors.
(a) What is the probability that a student chosen at random from
the class will receive a grade in the 70’s (i.e., between 70
and 80)?
(b) If a student is chosen at random from the class, what is the
student’s expected grade?
(c) A student is chosen at random from the class and you are
told that the student has a grade in the 70’s. What is the
probability that the student is a junior?
6.28 (†) Suppose X and Y are positive, independent continuous random variables
with probability density functions fX (x) and fY (y), respectively. Let Z =
X/Y .
(a) Express the probability density function for Z in terms of an
integral that involves only fX and fY .
(b) Now suppose that X and Y can be positive and/or negative.
Express the probability density function for Z in terms of
an integral that involves only fX and fY . Compare your
answer to the case where X and Y are both positive.
ANSWERS TO SELFTEST
EXERCISES
Chapter 6
S6.1 B S6.2 D S6.3 E S6.4 C S6.5 C
S6.6 B S6.7 C S6.8 C S6.9 C S6.10 B
S6.11 B S6.12 C S6.13 E S6.14 C S6.15 A
S6.16 C S6.17 A S6.18 D S6.19 E S6.20 D
S6.21 D S6.22 B
Remarks
S6.19 The correct answer is 1