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A series of problems and solutions related to probability theory and stochastic processes, including random processes, stationary random processes, poisson processes, and ergodic theory. The problems cover various topics such as autocorrelation, cross-correlation, power spectral density, and stability of stochastic systems. The solutions involve mathematical derivations and calculations using probability theory concepts. Useful for students studying probability theory, stochastic processes, and related fields.
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Problem solving session-
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Problem 17 Consider the vector random variable
given by
. It is given that
is normal with
mean vector
and correlation matrix
given by
and
We
t
t
2
1
2
3
now form the random process
Find the mean, autocorrelations and cross correlations of
( ) and
t
Y t
Y t
X t
t
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2
2
2
2
2
1
2
1
2
1 2
1
2
1
2
2
2
4
2
2
2
4
2
2
4
2
4
2
3
2
3
4
XX
X
t
t
t
t
t
t
t
t t
t
t
t t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
0
1
2
0
50
250 200 150 100
time
variance x(t)
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2
2
2
2
1
2
1
2
1 2
1
2
1
2
2
1
2
1
2
1
2
1
2
2
2
1
2
1
2 1
3 1
2
3 2
2
2
1
1
2
2
1
2
1
2
1 2
1
2
Check
XX
XX
XX
t
t
t
t
t t
t
t
t t
t
t
t
t
t
t
t t
t
t
t
Y t
Y t
Y t
t
t
t t
ok
t
t
t
t
t t
t
t
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2 2
2
2
2
exp
exp
exp
exp
0
exp
exp
exp
exp
XX
XX
X
t
a
j
t
a
j
t
a
j
t
j
X
t
X
t
a
j
t
j
t
a
j
R
a
S
a
j
d
a p
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2
2
XX
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UU
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2
2
exp
exp
exp
6
4
4
2
2
6
4
exp
exp
exp 2
exp
2
6
8cos
2 cos 2
6
8cos
2 cos 2
UU
YY
XX
XX
XX
XX
XX
YY
XX
XX
XX
XX
YY
R
i
d
S
R
a
i
d
i
a S
R
R
R
R
R
R
S
S
S
i
i
S
i
i
S
C
S
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Problem 20 Consider two independent random processes
and
which have zero mean and are stationary. Define
where
is a deterministic constant.
Determine PSD function of
t
t
Z t
t Y
t
Z t
XX
ZZ
XX
YY
ZZ
YY
Z t
t Y
t
Z t
t Y
t
t
t
Z t Z t
t Y
t
t
t
t
t
t
t
t
d
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0
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0
0
0
2
2
0
0
1
2
1
2
1
2
0 0
2
0
1
2
0
1
2
1
2
0 0
2
2
2
2
0
0
0
0
0
2
2
0
0
0
2
2
3
~
2
3
L
B
L L
B
L L L
B
F L
F
R
x
x dx
L
F L
F
R
x x
x
x
dx dx
L
F
x x I
x
x
dx dx
L
F
F
LI
x I dx
L
F L
F
LI
R
N
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0
0
0
0
0
0
0
2
2
2
0
0
0
0
2
2
0
0
0
L
A
L
L
A
L
A
A
B
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2
0
0
2
0
0
0
2
0
0
0
0
Similarly one can study
( )
2
2
2
1
( )
2
2
1
1
( )
2
2
Exercise: complete the characterization of
L
A
L
L
L
L
R L
L
L
M
M
x
f
x dx
L
L
M
L
x
f
x dx
x
f
x dx
L
L
L
x F
x
dx
x
f
x dx
M
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0
2
2
1
2
1
2
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