University of Illinois
Fall 2009
ECE 313:
Problem Set 7
Decision-Making Under Uncertainty
Due:
Wednesday October 14 at 4 p.m.
Reading:
Ross, Chapter 3; Powerpoint Lecture Slides, Sets 15-18
Decision-Making
and
Decision Making Under Uncertainty
: Additional reading
material available on the COMPASS web page for ECE 313
Noncredit Exercises:
Chapter 3:
Problems 53, 58, 59, 62, 63, 70-74, 78, 81
Theoretical Exercises 6, 7(a), 25, 26; Self-Test Problems 15-26.
Reminder:
Hour Exam I on Monday October 12, 7:00 p.m. – 8:00 p.m., 100 Noyes Lab.
Coverage of material is Problem Sets 1-6
1.
[A warm-up exercise]
If
H
0
is the true hypothesis, the random variable
X
takes on values 0, 1, 2, and 3 with probabilities
0.1, 0.2, 0.3, and 0.4 respectively. If
H
1
is the true hypothesis, the random variable
X
takes on values
0, 1, 2, and 3 with probabilities 0.4, 0.3, 0.2, and 0.1 respectively.
(a) Find the likelihood matrix
L
and indicate the
maximum-likelihood decision rule
by shading the
appropriate entries in
L
. What is the false-alarm probability
P
FA
and what is the missed-detection
probability
P
MD
for the maximum-likelihood decision rule?
(b) Now suppose that the hypotheses have
a priori
probabilities
π
0
= 0
.
7 and
π
1
= 0
.
3. Use the law
of total probability to ﬁnd the average error probability of the maximum-likelihood decision rule
that you found in part (a).
(c) Use the
a priori
probabilities given in part (b) to ﬁnd the joint probability matrix
J
and indicate
on it the Bayesian decision rule, which is also known as the minimum-error-probability (MEP)
or maximum
a posteriori
probability (MAP) decision rule. What is the average error probability
of the Bayesian decision rule? Is it smaller or larger than the average error probability of the
maximum-likelihood decision rule? In the latter case, provide a brief explanation as to why the