SHRI ANGALAMMAN COLLEGE OF ENGINEERING AND TECHNOLOGY

(An ISO 9001:2008 Certified Institution)

SIRUGANOOR, TIRUCHIRAPPALLI – 621 105

MA 1253- PROBABILITY AND RANDOM PROCESSES

UNIT-I RANDOM VARIABLES

PART-A

1. If -xke x>0

f(x)= 0 otherwise

is the pdf of a random variable X, find K.

2. Let X be a continuous random variable with p.d.f. 3x; 0<x<1

f(x)= 0 otherwise

find P(X≤0.6).

3. Find the value of c given the pdf of a randon variable X as 3

c1<x<

f(x)= x

0 otherwise

.

4. If the cdf of a RV is given by

2

0 <0

0 x<4f(x) = 16

1 4

for x

xfor

for x

And find P(x>1/x>3).

5. The cumulative distribution function of a random variable X is -x F(x) = 1-(1+x)e , x>0 Find the probability density function of X

6. The number of hardware failures of a computer system in a week of operations

has the following pmf:

No of failure 0 1 2 3 4 5 6

Probability .18 .28 .25 .18 .06 .04 .01

7. Is the function defined as follows a density function?

0 x<2

1(3 2x) 2 x 4

f(x) = 18

0 x>4

8. Define Moment generating function and write the formula to find mean and variance

9. Find the moment generating function of binomial distribution

10. The mean of a binomial distribution is 20 and standard deviation is 4. Find the

parameters of the didtribution.

11. Prove that the maximum value of the variance of the binomial distribution is n/4.

12. Find the moment generating function of Poisson distribution.

13. Find the mean of the poisson distribution which is approximately equivalent to B(300,0.2).

/2/

14. Define poisson distribution and state any two instances where poisson distribution may be

successfully employed

15. What is the area under the normal curve?

16. What is the characteristics of the Normal distribution.

17. Define exponential density function and write the formula to find mean and

variance.

18. Find the mean and variance of Geometric distribution.

19..Show that for the uniform distribution f(x)=1/2a,-a<x<a the moment generating

function about origin is sinhat/at.

20. A random variable Xhas pdf -xe 0

f(x)= 0 x<0.

x Find the density function of 1/X.

21. If Y=2X ,Where X is a Gaussian random variable with zero mean and variance 2 ,

find the probability density function of the random variable Y.

22. Given the random variable X with density function 2x; 0<x<1

f(x)= 0 otherwise

the probability density function of Y=83X .

23. If X is a Gaussian random variable with mean zero and variance 2 , find the

probability density function of Y= X .

PART-B

1. A random variable X has the following probability distribution

2 2 2

x : 0 1 2 3 4 5 6 7

P(x) :0 k 2k 2k 3k k 2k 7k k

Find (1) The value of k (2) P(1.5<X<4.5/X>2)

(3) The smallest value of for which P(X )>1/2 .

2. If the density function of a contimuous RV X is given by

ax 0 x 1

a 1 x 2f(x)=

3a-ax 2 x 3

0 otherwise

(1) Find the value of a

(2) Find the CDF of X (3) If x,y,z are 3 independent observations of X, what is

the probability that exactly one of these 3 is greater that 1.5?

3. A random variable X has the following probability distribution

x : 2 1 0 1 2 3

P(x) :0.1 k 0.2 2k 0.3 3kFind (1) The value of k

(2)Evaluate P(X<2) and P(-2<X<2) (3) Find the Cumulative distribution of X and

(4) Evaluate the mean of X.

/3/

4. The density function of a random variable X is given by

f(x) = KX(2-X) , 0 x 2 . Find K, mean,. Variance and rth moment.

5. .If X is a continuous R.V. with pdf by

2

x 0 x 1

3(x 1) 1 x 2

f(x)= 2

0 otherwise

Find (i) the value of a. (ii).the cumulative distribution of X.

(iii). If 1 2 3,x x andx are 3 independent observations of X.

W hat is the probability that exactly one of these 3 is greate than 1.5?

6. Find the moment generating function of binomial distribution hence find mean and variance

of the same.

7. Find the moment generating function of Poisson distribution and hence find its mean and

variance.

8. Find the mgf of the random variable with the probability law

x-1( x) = q p x = 1,2,3,........P X Find the Mean and Variance.

9. For the triangular distribution

x 0 x 1

(2 x) 1 x<2f(x) =

0 otherwise

find the mean variance and the MGF

10. If P(X=x)= , 1,2,3,4,5Kx x

0, otherwise

Represents a p.m.f. (i) Find K (ii) Find P(x being a prime number).

(iii)Find P 1 5

/ 12 2

x x iv) Find the distribution function.

11. The probability mass function of a Rv x is given by

P(i)= !

ic

i(i=0,1,2,…..) where λ>0 ; Find (i) P(x=0) (ii) P(x>2).

12. . The time (in hours) required to repairs a machine is exponential, distributed with

parameter λ = ½.

(i) What is the probability that the repair time exceeds 2 hours?

(ii) What is the conditional probability that a repair takes atleast 10h given that its

duration exceeds 9h?

13. If Xis a discrete random variable with probability function p(x)= 1

, 1,2,3......2x

x

Find the i) MGF (ii)Mean and variance.

14. The mileage which car owners get with a certain type of radial tyre is a random

variable having an exponential distribution with mean 40,000km find the probabilities

that one of these tyre’s will last (i) atleast 20,000km , (ii) atmost 30,000km.

15. Find the moment generating function of the Gamma distribution . Hence find its

mean and variance .

/4/

16.A random variable X is uniformly distributed in (-5,15). Find the probability function

of (i)Y= 2( ) .X ii Y X

17. Find the MGF of the random variable X having the pd.f. 2

xx>0

f(x)= 4

0 otherwise

x

e

Also find the first four moments about the origin.

18. Find the MGF ,Mean and variance of the distribution , whose probability function is

p(x)= , 0,1,2,......

0 otherwise

xq p x.

19.A wireless set is manufactured with 25 soldered joints each . On an average one joint

in 500 is defective .H ow many sets can be expected to be from defective joints in a

consignment of 10,000 sets?

20. Find the MGF of Normal distribution.

21. The daily consumption of bread in a hostel in excess of 2000 loaves is approximately

distributed as Gamma variable with parameters k=2, and 1

= .1000

The hostel has a

daily stock of 3000 loaves. What is the probability that the stock is insufficient on a

day?

UNIT-II TWO DIMENSIONAL RANDOM PROCESSES

PART-A

1.If -(x+y)f(x,y)= e , 0, 0x y is the joint pdf of x and y, find P(X<1)

2.If f(x,y)= 8xy,0 1,0x y x is the joint pdf of x and y, find f(y/x)

3.The regression equations of two variables x and y are x = 0.7y +5.2 and y = 0.3x + 2.8, find the

correlation coefficient between them.

4. State the Central Limit Theorem.

5. The regression equations of two variables x and y are 3x+ y = 10and 3x + 4y = 12, find the

correlation coefficient between them.

6. If the joint pdf of (X,Y) is given by f(x,y) = 2-x-y , 0 x< y 1, Find E(X)

7. Let X and Y be any two random variables and a,b be constants. Prove that

cov(aX,bY) = ab cov(X,Y)

8.The joint pdf of the RV (X,Y) is given by 2 2-(x +y )f(x,y)= ke , 0, 0x y .

Find the value of k and prove that X and Y are independent.

9. If the pdf of X is Xf ( ) 2 ,0 1x x x, find the pdf of Y = 3X +1.

10. Given the RV X with density function

2 ,0 1f(x,y)=

0,

x x

otherwise Find the pdf of38Y X

/5/

11.Define joint probability density function of two random variables X and Y

and state its properties. 12. Write the applications of central limit theorem.

13. Find the marginal density functions of X and Y if

2(2 5 ),0 , 1

f(x,y)= 5

0,

x y x y

otherwise

14.If X is a RV with cdf as F(X), show that the RV Y = F(X) is uniformly

distributed in (0,1)ss

15. If X and Y have joint pdf

( ),0 , 1f(x,y)=

0,

x y x y

otherwise Check whether X and Y are

independent.

16. The two regression lines are 4x + 5y +33 = 0, 20x +9y =107, find the

Correlation coefficient

17. The two regression lines are x + 6y = 4 ,2x + 3y =1. Find the mean

Values of x and y.

PART-B

1. If two dimensional random variables X and Y are uniformly distributed in

0≤ x ≤ y < 1 find

(i)Correlation coefficient rxy. (ii) Regression equations.

2. If the joint pdf of a two dimensional f( x, y ) is given by

f (x,y) = k (6-x-y) , 0 < x < 2, 2< y <4

0 , elsewhere

(i) find k (ii)P(x<1, y<3 )

(iii) P(x<1/y<3 ) (iv) P (y<3 /x< 1) (v)P(x + y < 3 )

3. Calculate the correlation coefficient for the following data:

Height of father (in inches) : 65 66 67 67 68 69 70 72

Height of son (in inches) : 67 68 65 68 72 72 69 71

4. The joint pdf of a two dimensional random variable ( X , Y ) is given by

f (x,y) = k (6-x-y) , 0 < x < 2, 2< y <4

0 , elsewhere

Find (i) the value of k (ii) P ( X < 1 / Y < 3 )

5. The joint pdf of the random variable ( X, Y ) is given by

f (x, y ) = Kxy e-(x2

+ y2

) , x>0 ,y>0

(i) Find K (ii) Prove that X and Y are independent.

6. Let X1,X2,……Xn be Poisson variates with parameter λ =2 . Let

S n = X1+X2+……+Xn , Where n = 75, find P (120 ≤ S n ≤ 160 ) using CLT.

/6/

7. State and prove central limit theorem.

8. Two random variables X and Y have the joint pdf

(4 ), 0 2,0 2

f(x,y)= 0 otherwise

k x y x y

Find the correlation coefficient between X and Y

9. Let X and Y be jointly distributed with pdf

1(1 ), 1, 1

f(x,y)= 4

0,

xy x y

otherwise .

Show that X and Y are not independent. 10. Let X and Y are normally distributed independent random variables with

Mean 0 and variance 2 . Find the pdf of R =2 2x y & = tan-1(Y/X).

11.Find the marginal and conditional densities if

f (x,y) = k ( 3x y +x3y ), 0 2,0 2x y .

12.The joint distribution of (X,Y) where X and Y are discrete is given in the

following table.

Verify whether X, Y are independent.

13. The joint pdf of a twodimensional RV (X,Y) is given by

2 ,0 1,0 2

f(x,y)= 3

0,

xyx x y

otherwise

Find (i) P(X>1 /2 ) (ii) P(Y<X ) (iii) P(Y<1/2 / X<1/2)

14. For 10 observations on price X and supply Y the following data were

Obtained: 2 2130, 220, 2288, 5506 3467X y x y and xy .

Obtain the Line of regression of Y on X and estimate the supply when

the price is 16 units.

X Y

0 1 2

0 0.1 0.04 0.06

1 0.2 0.08 0.12

2 0.2 0.08 0.12

/7/

15. The joint probability mass function of (X,Y) is given by

P(x,y) = k(2x +3y), x = 0,1,2 ; y = 1,2,3. Find all the marginal and Conditional

probability distributions. Also find the probability Distribution of ( X + Y ).

16. The lifetime of a certain brand of an electric bulb may be considered

a RV with mean 1200h and standard deviation 250h. Find the probability, using CLT,

that the average lifetime of 60 bulbs exceeds 1250h

17.The random variables X andY are statistically independent having a Gamma distribution

with b parameters (m,1/2)and (n,1/2) respectively. Derive the probability densityfunction of a

random variable U=X/(X+Y)

18.Find mean and variance of Gamma distribution and hence find mean and variance of

exponential distribution.

19.Find the correlation coefficient for the the following data:

X 10 14 18 22 26 30

Y 18 12 24 6 30 36

20.Given xy

( ),0 2,f (x,y)=

0,

cx x y x x y x

otherwise

(i) Evauate c (ii) Find Xf ( )x (iii) Y/Xf ( / )y x

(iv)f ( )Y y

21..From the following data ,find

(i) the two regression equations

(ii) the coefficient of correlation between the marks in Mathematics and

Statistics.

(iii) The most likely marks in Statistics when marks in Mathematics are 30.

22. The joint pdf of the random variable (X,Y) is given by

2 2-(x +y )f(x,y)=k xy e ;x>0,y>0 . Find the value of k. and prove also that

X and Y are independent.

23. If the independent random variables X and Y have the variances 36 and 16 respectively,

find the correlation coefficient between(X + Y) and (X-Y).

24. If the joint pdf of (X,Y) is given by f(x,y)=x +y,0 , 1,x y find the pdf of U = XY.

25. Given the joint pdf of (X,Y) as

8 ,0 1f(x,y)=

0,

xy x y

otherwise

Find the marginal and conditional pdfs of X and Y. Are X and Y independent?

Marks in

Mathematics

25 28 35 32 31 36 29 38 34 32

Marks in

Statistics

43 46 49 41 36 32 31 30 33 39

/8/

26.If y =2x-3 and y =5x+7 are the two regression lines, find the mean values

of x and y. Find the correlation coefficient between x and y. Find an

estimate of x when y = 1.

27.State and prove the central limit theorem for a sequence of independent

and identically distributed random variables.

28.If X and Y are independent variates uniformly distributed in (0,1), find the

distributions of XY and X/Y

29.Let X and Y be jointly distributed with p.d.f

1(1 ), 1, 1

f(x,y)= 4

0,

xy x y

otherwise

Show that X and Y are not independent but X2 and Y

2 are independent.

30. A sample of size 100 is taken from a population whose mean is 60 and Variance is 400. Using

C.L.T ; with what probability can we assert that the mean of the sample will not differ from

μ = 60 by more than 4.

31. The joint pdf of a two dimensional r.v (X,Y) is given by

2 2( )4 , 0, 0f(x,y)=

0,

x yxye x y

otherwise

Find the density function of U = 2 2U X Y

32. If the joint pdf of a r.v (X,Y) is given by

2 , 0 1,0 2f(x,y)= 3

0,

xyx x y

otherwise Find (a) P(X > 1/2) (b)P(Y<1) (c) P( Y >X)

(d)P(Y<1/2 /X<1/2) (e) Find the conditional density functions.

UNIT-III CLASSIFICATION OF RANDOM PROCESSES.

PART-A

1.Determine whether the given matrix is irreducible or not P =

0.3 0.7 0

0.1 0.4 0.5

0 0.2 0.8

/9/ 2.Show that the random process X(t) = A cos (ω 0 t + θ ) is not stationary,if A and

ω 0 are constants and θ is uniformly distributed in ( 0, π ).

3. Define a stationary process

4.Define Poisson process

5. Define ergodic random process

6.Define Wide-Sense Stationary process.

7. Define sine wave process

8. Define Wide-Sense Stationary process. Give an example.

9.Define Markov process

10. Define Strict Sense Stationary process and Wide Sense Stationary process.

11. Prove that sum of two independent Poisson process is also Poisson.

12. Prove that difference at two independent Poisson processes is not a Poisson process.

13. Define Strictly Sense Stationary (SSS) process.

14. Prove that a first order stationary random process has a constant mean.

15. Define Binomial process. Give an example for its sample function.

PART-B

1.If x(t) = A sin (ωt + θ ), where A and ω are constants and θ is RV uniformly distributed

over (-π , π ). Find the auto correlation of (y(t)), where

y(t) = x2(t).

1- <

f( )= 2

0 otherwise

2.For a random process x(t) = y sin ωt, y is an uniformly distributed random Variables in the

interval (-1,1) . Check whether the process is Wide Sense Stationary or not.

3. In a village road, buses cross a particular place at a Poisson rate of 4 per hour.

If a boy starts counting at 9.00 am.

(i) What is the probability that his count is 1 by 9.30 a.m ?

(ii) What is the probability that his count is 3 by 11.00 a.m?

(iii) What is the probability that his count is more than 5 by noon?

4.Suppose X(t) is a normal process with mean µ(t) = 3 and C(t1,t2) = 0.24 1 2e t t .

Find the probability that (8) (5) 1.X X

5.If X(t) is a random telegraph signal process with E { X(t)} = 0 and 2

( )R e .

Find mean and variance of the time average of X(t) over (-T,T).Is it mean Ergodic?

6. Show that the random process X(t) = A cos (ω 0 t + θ ) is WSS process, where A and ω 0 are

constants and θ is uniformly distributed random variable in ( 0, 2π ).

7.An office receives 3 calls per minute on an average. What is the probability of receiving

(i) no calls in a one-minute interval (ii) at most 3 calls in a 5 minutes interval

8. A radio active source emits particles at a rate of 5 per minute according to a Poisson process.

Each particle emitted has probability 0.6 of being recorded. Find the probability that 10

particles are recorded in a 4- minute period.

/10/

9. If {X(t)} is a Gaussian process with µ = 10 and C(t1,t2) = 1 2

16t t

e , find the

probability that (10) (6) 4.X X

10. Show that the random process X(t) = A cos ωt + B sin ωt is WSS; if A and B

are random variables such that

(i) E(A) = E(B) =0 (ii) E(A2) = E(B

2) =0 and (iii) E(AB) = 0

11. Classify the random process and explain with an example

12. Given a random variable Y with characteristic function ( ) and a random process

X(t) = cos ( )t Y . Show that {X(t)} is stationary in the wide sense if (1)= 0 and (2) =0 ..

13.Three boys X,Y,Z are throwing a ball to each other. X always throws the ball to Y and Y

always throws the ball to Z, but Z is just as likely throw the ball to Y as to X. Show that the

process is Markovian. Find the transition probability matrix and classify the states.

14.A machine goes out of order whenever a component fails. The failure of this part follows a

Poisson process with a mean rateof 1 per week. Find the probability that 2 weeks have elapsed

since last failure. If there are 5 spare parts of this component in an inventory and that the next

supply is not due in 10 weeks, find the probability that the machine will not be out of order in

the next 10 weeks.

15.If X(t) = Y cos ωt +Z sin ωt, where Y and Z are two independent normal RVs

with E(Y) = E(Z) =0, E(Y2) = E(Z

2) = 2 and ω is a constant , prove that{X(t)}

is a SSS process of order 2. 16. State the Postulates of a Poisson process and derive its probability law.

17.Prove that the sum of two independent Poisson process is a Poisson process.

18.Define random process. Classify it with an example.

19. The process {X(t)} whose probability distribution under certain conditions is

given by

1

1

( )n = 1,2,.....

(1 )P{X(t)}=

01

n

n

at

at

atn

at

Show that it is not stationary.

20. A man either drives a car (or) catches a train to go to office each day. He never goes

2 days in a row by train but if he drives one day, then the next day he is just as likely

to drive again as he is to travel by train. Now suppose that on the first day of the week,

the man tossed a fair die and drove to work if and only if a 6 appeared. Find

(i)the probability that he takes a train on the third day and (ii) the probability that he

drives to work in the long run.

21. For a random process X(t) = Y sin ωt , Y is an uniform random variable in the

Interval -1 to +1. Check whether the process is wide sense stationary or not.

22. The tpm of a matrix of a Markov chain {Xn}, three states 1,2 and 3 is

P =

0.1 0.5 0.4

0.6 0.2 0.2

0.3 0.4 0.3

and the intial distribution is P(0)

= (0.7, 0.2, 0.1).

Find (i) P{ X2 = 3} and (ii) P { X3 = 2, X2 =3, X1 =3, X0 =2 }

/11/

23.If {X(t)} and {Y(t)} are two independent Poisson processes, show that the

Conditional distribution {X(t)} given {X(t) + Y(t) } is binomial.

24. Show that the random process X(t) = cos ( t + Ф ) where Ф is a random

variable uniformly distributed in ( 0, 2π ) is

(i) first order stationary

(ii) stationary in the wide sense

(iii) ergodic (based on first order or second order averages)

25.Define Poisson process and obtain the probability distribution for that.Also Find the

autocorrelation function of the process.

26.Show that when events occur as a Poisson process, the time interval between

successive events follow exponential distribution.

UNIT – IV- CORRELATION AND SPECTRAL DENSITIES

PART – A

1.Find power spectral density of a stationary process whose auto correlation function is e 2.Find the auto correlation function of a stationary process whose power spectral density function

is given by

2 1s( )=

0 1

for

for

3.Define Cross – Correlation function.

4.What is meant by spectral analysis?

5.State Wiener – Khintchine relations.

6.The autocorrelation function of the random telegraph signal process is given by

22( )R e . Determine the power density spectrum of the random telegraph signal.

7.Given that the autocorrelation function for a stationary ergodic process with no periodic

components is 2

4( ) 25 .

1 6xxR Find the mean value and variance of the process ( )X t .

8.If the autocorrelation function of stationary process is 2

4( ) 36

1 3xxR

T

Find the mean and variance of the process

9.Find the power spectral density of a random signal with autocorrelation function T

e

10.The autocorrelation function of a stationary random process is

2

9( ) 16 .

1 6xxR Find the variance of the process.

11.Define power spectral density function of a stationary random process X(t).

12.Given the power spectral density: 2

1( )

4xxS ,find the average power of the process .

/12/

PART – B

1.Find the power spectral density of the random process ( )x t if ( ) 1E x t and

( ) 1xxR e .

2.If ( )x t is a WSS process with auto correlation function

( )XXR and if y(t) = x ( t+ ) - x (t - ) prove that syy( ) = 4sm2 ( ) sxx( )

3. Given that a process ( )X t has an auto correlation function ( )xxR Ae . 0cos( ),

Where A>0, >0 and 0 are real constants, find the power spectral of the process

4.Show that the power spectrum of a random process ( )x t is real and verify that

( ) ( )xx xxS S .

5.Find the auto correlation function of the process ( )x t for which the power density spectrum is

given by

21 1s ( ) =

0 1xx

for

for.

6. Find the output power density spectrum and output autocorrelation function for a system with

h(t) = e-t ;t 0 for an input with power density spectrum 0

2

h, f .

7. Find the power spectral density of a WSS process with auto correlation function 2

( ) .R e

8.Find the power spectral density of a random process whose auto correlation function is

( ) cosR e .

9.Consider two random processes ( ) 3cos( )X t t and ( ) 2cos( )Y t t

Where 2

and is uniformly distributed random variable over (0,2 ).

Verify whether ( ) (0) (0)xy xx yyR R R .

10.Define Power spectral density and cross spectral density of a random process. State their

properties.

11.If the power spectral density of a wide sense stationary process is given

( ),

s ( ) =

0

b

, find the autocorrelation function of the process.

12.Find the power spectral density of a WSS process with autocorrelation function.

2

( )R e .

13.The power spectral density function of a zero mean WSS process ( )X t is given by

01,s ( ) =

0 elsewhere Find R( ) and show that X(t) and

0

( )X t are uncorrelated.

14. The autocorrelation function of the random telegraph signal process is given by

22( ) .R e Determine the power density spectrum of the random telegraph signal .

/13/

15.State and prove Wiener-Khinchine theorem.

16. A random process ( )X t is given by X(t)=A cos pt + B sin pt,where A and B are idependent

RVS such that E(A) = E(B) = 0 and E(A2 )=E( B

2)= 2 . Find the power spectral density

of the process.

17. The power spectral density function of a zero mean wide – sense stationary process ( )X t is

given by 01,s ( ) =

0 elesewhere . Find R( ) and show also that X(t) and

0

( )X t are

uncorrelated.

18. If ( )X t is a band limited process such that s ( ) = 0xx ,when , prove that

2 22 (0) ( ) (0)xx xx xxR R R .

19.If Y(t) = A cos( 0 )t +N(t), where A is a constant, is a random variable with a uniform

distribution in ( , ) ( )and N t is a band limited Gaussian white noise with a power spectral

density

00,

2s ( ) =

0,

B

NN

Nfor

elesewherefind the power spectral density of ( )Y t . Assume

that N(t), and are independent.

20. The auto correlation function of the Poisson increment process is given

by

2

2

;

( ) = 1 ;

for

R . Prove that its spectral density is given by

2

2

2 2

4 sin ( /2)( ) 2 ( )S .

21. If ( )X t is a WSS process with auto correlation function ( )xxR T

( ) ( ) ( )

( ) 2 ( ) ( 2 ) ( 2 ).yy xx xx xx

Y t X t X t showthat

R T R T R T R T

22.Find the power spectral density of the random process, if its autocorrelation function is given by

cos

( )T T

xxR T e .

23.Show that if ( )X t is a WSS process then the output ( )Y t is a WSS process.

24.Given the power spectral density of a continuous process as 2

4 2

9( )

5 4xxS , find the mean square value of the process.

/14/

UNIT –V LINEAR SYSTEMS WITH RANDOM INPUTS.

PART-A

1.When do you a system is linear?

2.Define linear system.

3. State the properties of a linear filter.

4. If the input and output of the system Y(t) = ( ) ( )h u X t u du are WSS processes,

how are their power spectral densities related?

5. If {X(t)} and {Y(t)} in the system Y(t) = ( ) ( )h u X t u du are WSS processes,

how are their ACF’s related?

6. When is a system time-invariant ?

PART-B

1.If the input of a time-invariant stable linear system in a WSS process, then show

that the output will also be a WSS process.

2.Assume a random process {X(t)} is given as input to a system with the system

transfer function H ( ) = 1, - 0 < < 0 . If the auto-correlation function of the input process

is 0 ( )2

N, find the auto correlation of the output process.

3.For a linear system with random input X(t), the impulse response h(t) and output

Y(t) , obtain the power spectrum ( )yyS and cross power spectrum ( )xyS .

4.X(t) is the input voltage to a circuit (system) and Y(t) is the output voltage.

{X(t)} is a stationary random process with 0x and ( )xxR e

y, ( )yyS and ( )yyR , if the power transfer function is ( )

RH

R iLW

5. Show that in an input –out system the energy of a signal is equal to the energy of its

Spectrum.

6.A system has an impulse response ( ) ( )th r e u t , find the power spectral density

of the output y(t) corresponding to the input x(t).

7.Find out the out put power density spectrum and output autocorrelation function

for a system with ( ) , 0th t e t for an input with power density spectrum

0 ,2

f .