EE 278
October 28, 2009
Statistical Signal Processing
Handout #11
Sample Midterm Examination Problems
The following are old midterm problems. The midterm will cover lecture
notes 1–5, pages 1–6 of lecture notes 6, and homeworks 1–5, including the
Schwarz and Jensen inequalities and the extra problems. These problems
are meant for practice. The actual midterm will have fewer problems.
1.
Inequalities.
Label each of the following statements with =,
≤
,
≥
, or
None
. Label a statement
with = if equality always holds. Label a statement with
≥
or
≤
if the corresponding inequality
holds in general and strict inequality holds sometimes. If no such equality or inequality holds
in general, label the statement as
None
. Justify your answers.
a. P(
A
) vs. 1

(P(
A
c
, B
) + P(
A
c
, B
c
)).
b. E(
X
1
X
2

X
3
) vs. E(
X
1

X
3
) E(
X
2

X
3
) if
X
1
and
X
2
are independent.
c. E[Var(
X

Y, Z
)] vs. E[Var(
X

Y
)].
d. E[Var(
X

Y
)] vs. E[Var(
X

g
(
Y
))]. (Hint: use the result of part (c).)
e. E
Z
[E(
X
2

Z
) E(
Y
2

Z
)] vs. [E
Z
(Cov(
X, Y

Z
))]
2
.
f.
E
parenleftBig
log
2
parenleftBig
1 +
√
X
parenrightBigparenrightBig
vs. 1 if
X
≥
0 and E(
X
)
≤
1.
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 Fall '09
 BalajiPrabhakar
 Signal Processing, Probability theory, X1 X2 X3, MSE estimate

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