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Probability Theory and Stochastic Processes Problems, Slides of Stochastic Processes

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.

Typology: Slides

2012/2013

Uploaded on 04/24/2013

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Problem solving session-2
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Problem solving session-

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2

1

2

3

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

Y
Y
Y
Y
Y
Y
R
R
YY

2

1

2

3

now form the random process

Find the mean, autocorrelations and cross correlations of

( ) and

X

t

Y

Y t

Y t

X t

X

t

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4

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

X

t

t

t

R

t

t

t

t

t t

t

t

t t

X

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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5



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

R

t

t

t

t

t t

t

t

t t

X

t

X

t

R

t

t

t

t

t t

t

X

t

X

t

Y

Y t

Y t

Y

Y t

t

t

t t

ok

X

t

X

t

R

t

t

t t

t

t

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7

 

 

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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8

 

 

2

2

Problem 19: A random process

is given by

  • 2

where

( ) is a zero mean stationary random process

with PSD functionDetermine the PSD function of

XX

Y

t

Y

t

X

t

X

t

X

t

X t

C

S

Y t

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10

exp

Consider

exp

exp

exp

exp

exp

UU

S

R

i

d

R

a

i

d

R u

i

u

a

d

i

a

R u

i

u d

i

a S













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11

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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13

 

 

 

 

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

X

t

Y

t

Z t

X

t Y

t

Z t

 

 

 

 

 

 

 

 

XX

ZZ

XX

YY

ZZ

YY

Z t

X

t Y

t

Z t

X

t Y

t

X

t

Y

t

Z t Z t

X

t Y

t

X

t

Y

t

X

t

X

t

Y

t

Y

t

Y

t

R
R
R
S
S
S

d



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14

0

Problem 21A simply supported beam of span

carries a distributed

load

( ). The load is modeled as a segment of stationary

random process as

such that

0 and

. Determine the

L

f

x

f

x

F

x

x

x

x



 

following

Bending moment at midspanJoint pdf of reactions at the two supports.

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16

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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17

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

R L

L

x

f

x dx

R

L

x

f

x dx

L

x F

x

dx

L

L

F L

F

L

x

x dx

L

F L

F L

F

LI

R

R

F L

F

LI

R

N



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19

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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20

0

2

2

1

2

1

2

Problem 22A cantilever beam carries a randomly distributed load

as shown below. The load

( ) is modeled as

exp

Determine the bending moment at a sectionm

q x

q x

q

f

x

f

x

f

x

f

x

x

x

x

easured from the free end.

x

q x

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