University of Illinois
Spring 2010
ECE 313: Problem Set 7
Decision Making; Independent Events; System Reliability
Due:
Wednesday March 10 at 4 p.m.
Reading:
Ross, Chapter 3; PowerPoint Lecture Slides 1519
Noncredit Exercises: Chapter 3: Problems 53, 58, 59, 62, 63, 7074, 78, 81
Theoretical Exercises 6, 7(a), 25, 26; SelfTest Problems 1526.
This Problem Set contains seven problems.
1. [
Mutually exclusive events
] Let A and B denote two mutually exclusive events that can occur
on a trial of an experiment. Repeated independent trials of the experiment are carried out until
either the event A or the event B occurs. What is the probability that A occurs before B does?
(See Example 4h in Chapter 3 of Ross)
2. [
Detection problem with geometric distribution vs. Poisson distribution
] Consider a
detection problem with the following two hypotheses for an observation X:
H
0
: X has the Poisson distribution with parameter
λ
= 10:
H
1
: X has the geometric distribution with parameter p = 0:1:
(a) To get some intuition about this problem, calculate the mean, variance, and standard
deviation of X under H
0
and under H
1
:
(b) Describe the ML decision rule. Express it as directly in terms of X as possible.
(c) Describe the MAP decision rule, under the assumption that H
0
is a priori 5 times more
likely than H
1
(i.e,
π
0
/
π
1
= 5): Express the rule as directly in terms of X as possible.
3. [
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 Spring '08
 Milenkovic,O
 Probability theory

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