nctuee07f
Stochastic Processes
Midterm 1
1:00 pm – 3:20 pm, 10/23/07
IMPORTANT:
•
Remember to write down your id number and your name.
•
There are 4 problem sets with 110
points in total.
•
Please provide detailed explanations/reasonings with your answers. Correct answers
with
out
any explanations will carry
NO
credits.
On the other hand, wrong answers with
correct reasonings will get partial credits.
You may need the following formulas:
•
Let
x
= [
X
1
, X
2
,
· · ·
, X
n
] be a real Gaussian random vector with mean vector
m
x
and covariance matrix
K
x
. Then, the joint pdf
f
x
(
x
) is given by
f
x
(
x
) =
1
(2
π
)
n/
2
det(
K
x
)
1
/
2
exp

1
2
(
x

m
x
)
T
K

1
x
(
x

m
x
)
¶
.
•
The joint moment generating function for
N
random variables
X
1
· · ·
X
N
is
defined by
θ
(
t
1
, t
2
,
· · ·
, t
N
) =
E
"
exp
ˆ
N
X
i
=1
t
i
X
i
!#
.
•
The correlation coefficient
ρ
between random variables
X
and
Y
is defined by
ρ
,
E
£
(
X

μ
X
)(
Y

μ
Y
)
/
σ
X
σ
Y
,
where
μ
X
=
E
[
X
],
μ
Y
=
E
[
Y
],
σ
2
X
= Var(
X
), and
σ
2
Y
= Var(
Y
).
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 Winter '10
 GFung
 Variance, Probability theory, probability density function, Gaussian random variable, Gaussian random vector

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