University of Illinois
Fall 2009
ECE 313:
Problem Set 4
Counting Random Variables, MaximumLikelihood Estimation
Due:
Wednesday September 23 at 4 p.m.
.
Reading:
Ross, Chapter 4
Noncredit Exercises:
DO NOT turn these in.
Chapter 4:
Problems 32, 3839, 4043, 4752;
Theoretical Exercises 1619; SelfTest Problems 9, 13, 15, 16.
1.
[Graphical study of binomial pmfs]
Use a spreadsheet/Mathematica/MATLAB for this problem. Let
A
denote an event of probability
p
.
(a) For
p
= 0
.
1
,
0
.
25
,
0
.
4
,
0
.
5
,
0
.
6
,
0
.
75, and 0
.
9, ﬁnd the numerical values of the probabilities that
A
occurs 0
,
1
,
2
,...,
10 times on 10 trials.
(b) You have computed the pmf of a binomial random variable
X
p
with parameters (10
,p
) for seven
choices of
p
. For each value of
p
, draw a bar graph of the pmf of
X
p
. (The pmf of
X
0
.
5
is shown
on page 157 of the text!).
(c) What is the relationship between the pmfs of
X
p
and
X
1

p
?
2.
[The probability of an even number of successes]
Let
X
denote a binomial random variable with parameters (
N,p
). What is
P
{
X
is even
}
?
Hint: Expand (
x
+
y
)
n
+ (
x

y
)
n
using the binomial theorem and then set
x
= 1

p
,
y
=
p
.
3.
[The pricing of airline tickets]
Eight persons have purchased tickets ($
F
per person) for travel in a 5passenger plane on a scheduled
airline ﬂight in Ruritania. The number of persons who actually show up to travel can be modeled as a
binomial random variable
X
with parameters (8
,
0
.
5). Naturally, if more than 5 persons show up, only