University of Illinois Spring 1998
ECE313: Problem Set #9
Assigned: October 21, 1998
Due: October 28, 1998
Reading: Ross, Chapters 4 and 5
Additional problems: 4.20–4.23, 4.25, 4.27, 4.28, 4.32–4.40, 4.42, 4.43, 4.46, 4.48, 4.50, 4.51
1. Suppose that five boys and five girls are ranked according to their scores on an exam. Assume
that no two scores are alike, and that all the
10!
rankings are equally likely. Let
X
denote the
highest ranking achieved by a girl. For example, if the top-ranked person is a girl, then
X
=1
.
If the top-ranked girl is ranked second overall, then
X
=2
, and so forth.
(a) Find the probability mass function of
X
.
(b) Compute the expected value and variance of
X
.
2. A box contains 4 white balls and 4 black balls, and the following game is played. Each round of
the game consists of randomly selecting four balls from the box. If exactly two of them are white,
the game is over. Otherwise, the four balls are placed back in the box, so that it again contains 4
white balls and 4 black balls before the next round is performed. The rounds are repeated until
exactly two of the chosen balls are white.
(a) Let
X
denote the number of rounds in this game. Find the expected value and the variance
of
X
.
(b) When the game is over, we get a reward
Y
. The value of
Y
decreases exponentially with
the the number of rounds played. Specifically, if the game terminates after
n
rounds, then
Y
=1
=e
n
. Find the expected value of the reward
Y
.
3. Suppose that the continuous random variable
X
has probability density function
f
X
(
u
)
defined
as follows:
f
X
(
u
)=
c
(1
,
u
)
2
for
0
<u<
1
,and
f
X
(
u
)=0
elsewhere.
(a) Find the value of the constant
c
.
(b) Find the mean and the variance of
X
.
(c) Compute
P
(6
X
2
>
5
X
+1)
and
P
(6
X
2
>
7
X
,
2)
.
4. Let
X
denote a Poisson random variable with parameter
=0
:
5
.
(a) Compute
E
[
X
!]
.
(b) Show that
E
[
X
n
]=0
:
5
E
[(
X
+1)
n
,
1
]
.
(c) Use this result to compute
E
[
X
3
]
.
