nctuee07f
Stochastic Processes
Midterm 1
Solutions
1.
True or False
(6 points
×
5 = 30 points)
Please provide an explanation, a simple proof or a counter example, to your an
swer.
No points will be credited if answers are with true or false only
.
Try to be concise and to the point.
(a)
True
. This is just the singular value decomposition applied to a square
matrix.
(b)
True
.
For Hermitian matrix
A
, we know
A
=
A
H
.
The eigenvector
x
associated with the eigenvalue
λ
satisfies
Ax
=
λ
x
.
It follows that
λ
·
x
H
x
=
x
H
·
λ
x
=
x
H
Ax
=
x
H
A
H
x
since
A
=
A
H
=
Ax
H
x
=
λ
*
·
x
H
x
,
from which we know
λ
=
λ
*
. Thus, the eigenvalue of a Hermitian matrix
is always real.
(c)
False
. We need to add that
X
and
Y
are
jointly
Gaussian in order to
make such conclusion.
(d)
True
. See HW1 solutions.
(e)
True
.
Consider independent discrete random variables
X
and
Y
.
By
independence we know
p
X,Y
(
x, y
) =
p
X
(
x
)
p
Y
(
y
)
.
Then,
E
[
XY
]
=
X
x
X
y
xyp
X,Y
(
x, y
)
=
X
x
X
y
xyp
X
(
x
)
p
Y
(
y
) =
X
x
xp
X
(
x
)
·
X
y
yp
Y
(
y
)
=
E
[
X
]
E
[
Y
]
.
Thus,
X
and
Y
are uncorrelated.
1
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2.
Central Limit Theorem
(10 points)
Let
Y
=
N
X
i
=1
X
i
.
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 Winter '10
 GFung
 Normal Distribution, Variance, Probability theory, probability density function, Gaussian random variable

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