1
National Chiao Tung University
Department of Electronics Engineering
Stochastic Processes, Final Exam
Solutions
NOTE:
•
Remember to write down your name.
•
There are 5 problem sets with 100 points in total.
1.
True or False
(5+5+5+5=20 points)
Indicate whether the statement in each of the following is True or False. Please
provide an explanation, either a concise proof or a counter example, to your
answer. Otherwise, your answer will carry no credits.
(a) True. We can prove this using Chebyshev’s inequality (p. 380 in textbook).
(b) True. By the definition of independent increments, for all
t
1
, t
2
, . . . , t
N
such
that
t
1
< t
2
< . . . < t
N
, we have
X
(
t
1
)
, X
(
t
2
)

X
(
t
1
)
, . . . , X
(
t
N
)

X
(
t
N

1
)
are mutually independent for every integer
N >
1. If we choose
N
= 3,
and the time points
t
1
< s < t
2
, it is clear that
X
(
t
2
)

X
(
s
),
X
(
s
)

X
(
t
1
)
and
X
(
t
1
) are mutually independent.
(c) True. Since
X
(
t
) has
E
[

X
(
t
+
T
)

X
(
t
)

2
] = 0 for all
t
, we know
R
XX
(0) =
R
XX
(
T
). The CauchySchwarz inequality tells that
‡
E
h‡
X
(
t
+
T
+
τ
)

X
(
t
+
τ
)
·
X
(
τ
)
i·
2
≤
E
•
‡
X
(
t
+
T
+
τ
)

X
(
t
+
τ
)
·
2
‚
E
h
X
2
(
τ
)
i
.
The righthand side of the above is zero. And the lefthand side is
(
R
XX
(
t
+
T
)

R
XX
(
t
))
2
,
which is greater than zero due to the square. It follows the lefthand side
is zero, yielding
R
XX
(
t
+
T
) =
R
XX
(
t
)
.
(d) True. The autocorrelation function is
R
XX
(
τ
) =
E
[
x
(
t
+
τ
)
x
(
t
)] =
E
[
x
(
t
)
x
(
t
+
τ
)] =
R
XX
(

τ
)
.
So autocorrelation function is even, which can be used to
show that
S
XX
(
ω
) is also even.
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2. (5+5+5+5=20 points)
Answer the following questions. Please be concise and precise.
(a) See page 512 for the definition of ergodic in the mean, or page 514 Theo
reme 8.41 in textbook.
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 Winter '10
 GFung
 Variance, Electronics Engineering, Probability theory, probability density function, Stationary process

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