University of Illinois Fall 1998
ECE313: Problem Set #11
Assigned: Friday, November 6
Due: Friday, November 13
Reading: Ross, sections 1,2,3 of Chapter 6
Recommended problems : These exercises are from Ross: pp. 291–294, 1–5, 10, 11–13, 21;
pp. 296–297, 1–10, 13–19.
1. Suppose that two cards are drawn at random from a deck of 52 cards. Let
X
be the number of
aces obtained and let
Y
be the number of queens obtained.
(a) Find the joint probability mass function of
X
and
Y
.
(b) Find the marginal probability mass function of
X
and that of
Y
.
(For those who never play card games: there are 4 aces and 4 queens in a deck of 52 cards.)
2. The discrete random variables
X
and
Y
have joint p.m.f.
p
X;Y
(
u; v
)
given by
v
#
=u
!
0135
,
1
1
12
1
6
1
12
0
3
1
6
1
12
0
1
12
4 0
1
12
1
6
1
12
(a) Find the joint CDF of
X
and
Y
. Specify the value of
F
X;Y
(
u; v
)
for all
u
and
v
.
(b) Find the marginal probability mass functions
p
X
(
u
)
and
p
Y
(
v
)
of
X
and
Y
.
(c) Find
P
f
X
Y
g
and
P
f
X
+
Y
8
g
3. Is the following function
F
(
u; v
)=
(
0
u
+
v<
1
1
u
+
v
1
a valid joint CDF. Why or why not? Prove your answer and show your work.
4. The random variables
X
and
Y
have joint probability density function
f
X;Y
(
u; v
)=
(
2
e
,
(
u
+
v
)
0
<u<v<
1
0otherwise
(a) Sketch the
uv
plane with an indication of the region where
f
X;Y
(
u; v
)
is nonzero.
(b) Find the joint CDF of
X
and
Y
. Specify the value of
F
X;Y
(
u; v
)
for all
u
and
v
.
(c) Find the marginal CDFsof
X
and
Y
by setting, respectively,
v
and
u
to
+
1
in the answer
to part (b).
