ECE 5620
Spring 2011
Homework #2: Issued Feb. 2, Due Feb. 18 at 5 p.m.
Complete all ten problems.
1. Problem 16 in Chapter 2 of the text
2. Problem 25 in Chapter 2 of the text
3. Problem 29 in Chapter 2 of the text
4. Problem 12 in Chapter 3 of the text
5. Problem 8 in Chapter 7 of the text
6. Problem 9 in Chapter 7 of the text
7. Suppose that
X
1
,...,X
n
are independent.
(a) Show that
I
(
X
1
,...,X
n
;
Y
)
≥
n
X
i
=1
I
(
X
i
;
Y
)
.
(b) Give a collection of Markov chain conditions that is necessary and suf
ﬁcient for equality in (a).
8. Show that
H
(
Y

X
)
is a concave function of the joint PMF
p
(
x,y
)
. Is it
strictly concave?
9. Let
X
1
,...,X
n
be i.i.d. Bernoulli(
p
) and let
Y
1
,...,Y
n
be i.i.d. Bernoulli(
q
).
Show that

1
n
log
p
X
(
Y
1
,...,Y
n
)
converges in probability and determine the limit in terms of
p
and
q
.
10.
Typical vs. Expected Behavior
The Acahti Trust Company offers two investment instruments, the Lousy
Money Market Account and the Risky Mutual Fund. After one year, the
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 '08
 WAGNER
 money market account, Risky Mutual Fund, typical growth rate

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