ISyE 2027: Probability with Applications
Handout 18
Sigr´un Andrad´ottir
October 28, 2021
Assignment 8
Due November 4, 2021, 11 PM
Remember to attach a fully completed cover sheet to your homework
Problem 1
(6 points)
Do Problem 4 on Assignment 7.
Problem 2
(17 points)
a.
(2 points) Consider discrete random variables
X
and
Y
whose joint probability mass
function (PMF)
p
X,Y
(
a, b
) is provided in the following table:
p
X,Y
(
a, b
)
b
= 5
b
= 10
a
= 1
0
.
15
p
a
= 2
0
.
45
0
.
05
(
p
X,Y
(
a, b
) = 0 if
a
6∈ {
1
,
2
}
or
b
6∈ {
5
,
10
}
). Determine the value of
p
(if possible).
Explain.
b.
(2 points) Determine the joint cumulative distribution function (CDF)
F
X,Y
of the
random variables
X
and
Y
defined in part
a
.
Make sure to fully specify this
function. Explain.
c.
(5 points) Do Problem 9.1, page 127 in the text.
Also, compute the marginal
cumulative distribution functions (CDFs) of
X
and
Y
. Make sure to fully specify
the functions. Are
X
and
Y
independent? Explain.
d.
(4 points) Do Problem 9.3, page 129 in the text (recall that
{
1
,
4
} × {
1
,
4
}
=
{
(1
,
1)
,
(1
,
4)
,
(4
,
1)
,
(4
,
4)
}
). Explain.
e.
(4 points) Suppose 0
< p, q, r <
1 and consider the random variables
Z
1
,
Z
2
, and
Z
3
defined as follows:
Z
1
=
(
1
with probability
p,
2
with probability 1

p
;
Z
2
=
(
5
with probability
q,
10
with probability 1

q
;
Z
3
=
(
10
with probability
r,
20
with probability 1

r.
Assume that the random variables
Z
1
,
Z
2
, and
Z
3
are independent. Define the
random variables
X
=
Z
1
+
Z
2
and
Y
=
Z
1
+
Z
3
. Specify the joint probability
mass function (PMF) of
X
and
Y
. Make sure to fully specify this function. Are
X
and
Y
independent? Explain.
1