Probability Random Process QB Mahalakshmi Engg

MA 2261 Probability and Random process IV Sem ECE – S.SARULATHA Asst.Prof./MATHS

1

MAHALAKSHMI ENGINEERING COLLEGE

TIRUCHIRAPALLI – 621213

QUESTION BANK

DEPARTMENT: ECE SEMESTER: IV

SUBJECT CODE / Name: MA 2261/PROBABILITY AND RANDOM PROCESS

UNIT – I: RANDOM VARIABLES

PART -A (2 Marks)

1. The CDF of a continuous random variable is given by

5

0 , 0,( ) ,

1 ,0x

xF x X

e x

Find the pdf and meanof

(AUC Nov/Dec 2011)

(AUC Apr/May 2011)

2. The probability that a man shooting a target is 1/ 4 . How many times must he fire so

that the probability of his hitting the target atleast once is more than 2 / 3?

(AUC May/Jun 2012)

3. Find C, if 2

; 1,2,....3

n

P X n c n

(AUC May/Jun 2012)

4. A continuous random variable X has probability density function

23 ,0 1( )

0 ,

x xf x

otherwise

.Find K such that 0.5P X k

(AUC Nov/Dec 2010)

5. If X is uniformly distributed in ( , )2 2

. Find the probability distribution function of

𝑌 = 𝑡𝑎𝑛 𝑋 (AUC Nov/Dec 2010)

6. Establish the memory less property of the exponential distribution.

(AUC Apr/May 2011)

7. If the probability density function of a random variable X if

( ) 0 2, ( 1.5 / 1).2

xf x in x P X X find (AUC Apr/May 2010)


2

8. If the MGF of a uniform distribution for a random variable 1 5 4 . ( ).X is e e E xt

t tfind

(AUC Apr/May 2010)

9. If x is a normal random variable with mean zero and variance 2 , find the PDF of

Y e x (AUC Nov/Dec 2011)

10. The moment generating function of a random variable X is given by

3( 1)( ) . [ 0]?teM t e P X Whatis (AUC Nov/Dec 2012)

11. An experiment succeeds twice as often as it fails. Find the chance that in the next 4

trials, there shall be at least one success. (AUC Nov/Dec 2012)

PART –B (16 Marks)

1. The probability function of an infinite discrete distribution is given by

1

( 1,2,3....)2

P X x x x Find

(i) The value of X

(ii) P[X is even]

(iii) P[X is divisible by 3] (AUC Nov/Dec 2011)

2. A continuous random variable X has the PDF 2,

( ) 1

0 ,

kx

f x x

otherwise

Find a) the value of k

b) Distribution function of X

c) 0P X (AUC Nov/Dec 2011)

3. Let X and Y be independent normal variates with mean 45 and 44 and standard

deviation 2 and 1.5 respectively. What is the probability that randomly chosen values

of X and Y differ by 1.5 or more? (AUC Nov/Dec 2011)

4. If X is a uniform random variable in the interval (-2,2) find the probability density

function ( )Y X and E Y

(AUC Nov/Dec 2011)

5. A random variable X has the following probability distribution

2 2 2 2

0 1 2 3 4 5 6 7

( ) 0 2 2 3 2 7

x

p x k k k k k k k k

Find, (i) The value of k

(ii) (1.5 4.5 / 2)P X X and (iii) The smallest value of n for which


3

1( )

2P X n (AUC Nov/Dec 2010)

(AUC May/Jun 2012)

6. Find the MGF of the random variable X having the probability density function

3, 0

( ) 4

0 ,

e xf x

-x2

elsewhere

Also deduce the first four moments about the origin.

(AUC Nov/Dec 2010)

7. If X is the uniformly distributed in (-1,1),then find the probability density function of

sin2

xY

(AUC Nov/Dec 2010)

8. If X and Y are independent random variables following

(8,2) (12,4 3)

2 2 2

N N

P X Y P X Y

and respectively,findthevalueof suchthat(AUC Nov/Dec 2010)

9. The probability mass function of random variable X is defined as

2 2( 0) 3 , ( 1) 4 10 , ( 2) 5 1, 0,

( ) 0 0,1,2.

( )

( ) (0 2 / 0)

1( ) ( )

2

P X c P X c c P X c Wherec and

P X r if r

i

ii P X X

iii X F X

)Find

The valueof c.

Thelargest valueof for which

(AUCApr / May2010

10. If the probability that an applicant for a driver’s license will pass the road test on any

given trial is 0.8. What is the probability that he will finally pass the test

(i) On the fourth trial and

(ii) In less than 4 trials? (AUC Apr/May2010)

11. The marks obtained by a number of students in a certain subject are assumed to be

normally distributed with mean 65 and standard deviation 5. If 3 students are

selected at random from this group, what is the probability that atleast one of them

would have scored above 75? (AUC Apr/May2010)

(AUC Apr/May 2011)

12. The Probability distribution function of a random variable X is given by

,0 1

( ) (2 ) ,1 2

0 ,

, (0.2 1.2)

(0.5 1.5 / 1)

( ).

x x

f x k x x

k P x

P x X

f x

X

(i)Findthe valueof (ii)Find

(iii)What is

(iv)Findthedistribution functionof

otherwise (AUC Apr/May 2011)


4

13. Derive the mgf of Poisson distribution and hence or otherwise deduce its mean and

variance. (AUC Apr/May 2011)

14. Find the M.G.F of the random variable X having the probability density function

, 0( ) 4 2

0 ,

xe x

f x

elsewhere

x

. Also deduce the first four moments about the

origin. (AUC May/Jun 2012)

15. Given that X is distributed normally, if ( 45) 0.31 ( 64) 0.08P X and P X , find

the mean and standard deviation of the distribution. (AUC May/Jun 2012)

16. The time in hours required to repair a machine is exponentially distributed with

parameter 1

2

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

(2) What is the conditional probability that a repair takes atleast 10 hours

given that its duration exceeds 9 hours? (AUC May/Jun 2012)

17. If the probability density of X is given by 2(1 ) 0 1,

( )0

x for xf x

otherwise

find its rth

Moment. Hence evaluate E[(2X+1)2]. (AUC Nov/Dec 2012)

18. Find MGF corresponding to the distribution 2

1, 0

( ) 2

0

ef

otherwise

and hence find

its mean and variance. (AUC Nov/Dec 2012)

19. Show that for the probability function

1, 1,2,3...

( 1)( ) ( ) ( ) .

0

xx xp x P X x E X

otherwise

doesnotexist

(AUC Nov/Dec 2012)

20. Assume that the reduction of a person’s oxygen consumption during a period of

Transcendental Meditation (T. M) is a continuous random variable X normally

distributed with mean 37.6 cc/mm and S. D 4.6 cc/min. Determine the probability that

during a period of T.M a person’s oxygen consumption will be reduced by

(1) At least 44.5 cc/min

(2) Utmost 35.0 cc/min

(3) Anywhere from 30.0 to 40.0 cc/mm. (AUC Nov/Dec 2012)


5

21. The random variable X has exponential distribution with

,0( ) ( ) ( )

0 ,

xe xf x f X f x

otherwise

Find the density function of the variable given

by (1) Y=3X+5 , (2) Y=X2 (AUC Nov/Dec 2012)


6

UNIT – II: TWO DIMENSIONAL RANDOM VARIABLES

PART -A (2 Marks)

1. If the joint probability distribution function of (X,Y) is

( ) , 0, 0( , )

0 ,

x y

XY

e x yf x y

otherwise

Check whether X and Y are independent.

(AUC Nov/Dec 2011)

2. The regression equations are 3 2 26 6 31.x y and x y Find the correlation

coefficient between X and Y.

(AUC Nov/Dec 2011)

3. Let X and Y be continuous random variables with joint pdf

( )( , ) ,0 2, ( , ) 0

8XY XY

x x yf x y x x y xand f x y

elsewhere. Find ( / )Y

X

f y x

(AUC Nov/Dec 2010)

(AUC Apr/May 2011)

4. State Central Limit Theorem for iid random variables. (AUC Nov/Dec 2010)

(AUC May/Jun 2012)

5. Find the acute angle between the two lines of regression, assuming the two lines of

regression. (AUC Nov/Dec 2010)

(AUC Nov/Dec 2012)

6. Find the value of k, if ( , ) (1 )(1 ) 0 4,1 5 ( , ) 0f x y k x y in x y and f x y

otherwise, is to be the joint density function. (AUC Apr/May 2010)

7. Define wide-sense stationary random process. (AUC Apr/May 2010)

(AUC May/Jun 2012)

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

26( ) ,0 1, 0 1

( , ) 5

0 ,

x y x yf x y

otherwise

(AUC Nov/Dec 2012)


1. The joint pdf of random variable X and Y is given by

8,1 2

( , ) 9

0 ,

xyx y

f x y

otherwise

.

Find the conditional density functions of X and Y. (AUC Nov/Dec 2011)

(AUC Apr/May 2010)

2. The joint pdf of the two dimensional random variable (X,Y) is


7

2 ,0 1,0 1( , )

0 ,

x y x yf x y

otherwise

. Find the correlation coefficient between X

and Y. (AUC Nov/Dec 2011)

3. If 1 2 3, , ,...X X X Xn are uniform variates with mean=2.5 and variance=3

4, use Central

limit theorem to estimate 1 2 3(108 12.6) ... . 48.P S WhereS X X X X n n n n

(AUC Nov/Dec 2011)

4. If X and Y are independent with a common pdf(exponential) , 0

( )0 , 0

e xf x

x

x

and

, 0( )

0 , 0

e yf y

y

y

. Find the PDF for X-Y. (AUC Nov/Dec 2011)

5. Two random variables X and Y have the joint pdf given by

2(1 ) ,0 1,0 1( , )

0 ,XY

k x y x yf x y

otherwise

. Find

(i) The value of k,

(ii) Obtain the marginal probability density functions of X and Y

(iii) Also find the correlation coefficient between X and Y.

(AUC Nov/Dec 2010)

6. If X and Y are independent continuous random variables. Show that the pdf of

𝑈 = 𝑋 + 𝑌 is given by ℎ(𝑢) = ( ) ( )X Yf u f u v dv

. (AUC Nov/Dec 2010)

7. If Vi , i=1,2,3,…20 are independent noise voltages received in an adder and V is the

sum of the voltages received, find the probability that the total incoming voltage V

exceeds 105, using the central limit theorem. Assume that each of the random

variables Vi is uniformly distributed over (0, 10). (AUC Nov/Dec 2010)

8. If X and Y are independent Poisson random variables with respective parameters

1 2.and Calculate the conditional distribution of X, given that 𝑋 + 𝑌 = 𝑛.

(AUC Apr/May 2011)

9. The regression equation of X and Y is 3 5 108 0.y x If the mean value of Y is 44

and the variance of X is 9

16th of the variance of Y. Find the mean value of X and the

correlation coefficient. (AUC Apr/May 2011)

10. If X and Y are independent random variables with density function,

1, 1 2 ,2 4( ) , ( ) 6

0,0 ,

X Y

yx y

f x f y

otherwise

otherwise

, find the density function of

Z=XY. (AUC Apr/May 2011)


8

11. Find the marginal distributions, conditional distribution of X given Y=1 and conditional

distribution of Y given X=0. (AUC Apr/May 2010)

12. Find the covariance of X and Y, if the random variable (X,Y) has the joint pdf

( , ) ,0 1, 0 1f x y x y x y and 𝑓(𝑥, 𝑦) = 0,otherwise.( AUC Apr/May 2010)

13. A sample of size 100 is taken from a population whose mean is 60 and variance is

400. Using central limit theorem, find the probability with which the mean of the

sample will not differ from 60 by more than 4. (AUC Apr/May 2010)

14. For the bivariate probability distribution of (X,Y) given below

Y

X

1 2 3 4 5 6

0 0 0 1/32 2/32 2/32 3/32

1 1/16 1/16 1/8 1/8 1/8 1/8

2 1/32 1/32 1/64 1/64 0 2/64

Find the marginal distributions, conditional distribution of X given Y=1 and conditional

distribution of Y given X=0. (AUC Apr/May 2010)

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

( )( , ) , 0, 0.

x yf x y Kxye x y

2 2Find the value of K and Cov(X,Y). Are X and Y

independent?(AUC May/Jun 2012)

16. If x and Y are uncorrelated random variables with variances 16 and 9. Find the

correlation co-efficient between X+Y and X-Y. (AUC May/Jun 2012)

17. If x and Y are uncorrelated random variables with variances 36 and 16. Find the

correlation co-efficient between (X+Y) and (X-Y). (AUC Nov/Dec 2012)

18. Let (X, Y) be a two dimensional random variable and the pdf be given by

( , ) ,0 , 1.f x y x y x y Find the pdf of U=XY. (AUC May/Jun 2012)

19. Let 1 2 3, , ,...,X X X Xn be Poisson variates with parameter λ=2 and

1 2 3 ...S X X X X n n where n=75. Find 120 160p S n using central limit

theorem. (AUC May/Jun 2012)


9

UNIT – III: CLASSIFICATION OF RANDOM PROCESSES

PART -A (2 Marks)

1. What is random process said to be mean ergodic? (AUC Nov/Dec 2011)

2. If ( )X t is a normal process with 1 2( ) 10 ( , ) 16t and C t t e

t1t2 find the variance

of 𝑋 10 − 𝑋 6 . (AUC Nov/Dec 2011)

(AUC May/Jun 2012)

3. State the Postulates of a Poisson process. (AUC Nov/Dec 2010)

(AUC Apr/May 2011)

4. Consider the random process 𝑋(𝑡) = 𝑐𝑜𝑠(𝑡 + Ф) where Ф is a random variable with

density function 1

( ) ,2 2

f

. Check whether or not the process is wide

sense stationary. (AUC Nov/Dec 2010)

5. Prove the first order stationary process. (AUC Apr/May 2011)

6. Define a wide sense stationary process. (AUC Apr/May 2010)

7. Define a Markov chain and give an example. (AUC Apr/May 2010)

8. The autocorrelation function of a stationary random process is 2

9( ) 16

1 6R

.

Find the mean and variance of the process. (AUC May /Jun 2012)


1. Show that the random process ( ) cos( )X t A t is wide- sense stationary, if A

and are constants and θ is a uniformly distributed in (0,2π).(AUC Nov/Dec 2011)

2. The process ( )X t whose probability distribution under certain condition is given by

1

1

( ), 1,2,...

(1 )( )

, 01

atn

atP X t n

atn

at

n

n

Find the mean and variance of the process. Is

the process first-order stationary? (AUC Nov/Dec 2011)

(AUC Nov/Dec 2011)

(AUC Nov/Dec 2012)

3. State the Postulates of a Poisson process and derive the probability distribution. Also

prove that the sum of two independent Poisson processes is a Poisson process.

(AUC Nov/Dec 2011)

4. If the WSS process ( )X t is given by ( ) 10cos(100 ),X t t where is uniformly

distributed over ( , ), Prove that ( )X t is correlation ergodic.

(AUC Nov/Dec 2010)

(AUC May/Jun 2012)

(AUC Nov/Dec 2012)


10

5. If the process ( ) : 0X t t is a Poisson process with parameter λ, obtain

( ) .P X t n Is the process first order stationary? (AUC Nov/Dec 2010)

(AUC Nov/Dec 2012)

6. Prove that a random telegraph signal process ( ) ( )Y t X t is a wide sense

stationary process when ∝ is a random variable which is independent of 𝑋(𝑡),

assume that values -1 and +1 with equal probability and 1 2

2

( , )XXR t t e

t -t1 2

.

(AUC Nov/Dec 2012)

(AUC Nov/Dec 2010)

7. A random process 𝑋(𝑡) defined by 𝑋(𝑡) = 𝑈𝑐𝑜𝑠𝑡 + 𝑉𝑠𝑖𝑛𝑡 ,-∞< t <∞ , where A and B

are independent random variables each of which takes a value -2 with probability 1/3

and a value 1 with probability 2/3. Show that 𝑋(𝑡) is wide-sense stationary.

(AUC Apr/May 2011)

8. A random process has sample functions of the form 𝑋(𝑡) = 𝐴𝑐𝑜𝑠 (𝜔𝑡 + 𝜃) where ω is

constant. A is a random variable with mean zero and variance one and θ is a random

variable that is uniformly distributed between 0 and 2π. Assume that the random

variable A and θ are independent. Is 𝑋(𝑡) is a mean ergodic process?

(AUC Apr/May 2011)

9. If ( )X t is Gaussian process with μ(t)=10 and C(1 2,t t )=16 e

t1t2 , find the

probability that

(i) X(10)≤8

(ii) (10) (16) 4X X . (AUC Apr/May 2011)

10. Prove that the interval between two successive occurrences of a Poisson process

with parameter λ has an exponential distribution with mean 1/λ.

(AUC Apr/May 2011)

11. Examine whether the random process 𝑋(𝑡) = 𝐴𝑐𝑜𝑠 (𝜔𝑡 + 𝜃) is a wide sense

stationary random variable in ( , ). (AUC Apr/May 2010)

12. Assume that the number of messages input to a communication channel in an

interval of duration t seconds, is a Poisson process with mean λ=0.3. Compute,

(1) The probability that exactly 3 messages will arrive during 10 seconds

interval.

(2) The probability that the number of message arrivals in an interval of

duration 5 seconds is between 3 and 7. (AUC Apr/May 2010)

13. The random binary transmission process ( )X t is a wide sense process with 0

mean and auto correlation function ( ) 1RT

, where T is a constant. Find the

mean and variance of the time average of ( )X t mean-ergodic?

(AUC Apr/May 2010)

14. The transition probability matrix of a Markov chain X n , n=1,2,3,… having 3 states

1,2 &3 is

0.1 0.5 0.4

0.6 0.2 0.2

0.3 0.4 0.3

P

and the initial distribution is

(0)

2 3 2 1 0(0.7,0.2,0.1) ( ) 3 ( ) 2, 3, 3, 2P find i P X and ii P X X X X .

(AUC Apr/May 2010)


11

15. If ( )X t is a WSS process with autocorrelation ( )R =Ae

, determine the second

order moment of the RV (8) (5)X X . (AUC May / Jun 2012)

16. If the customers arrive at a counter in accordance with a Poisson process with a

mean rate of 2 per minute, find the probability that the interval between 2 consecutive

arrivals is (1) more than 1 minute

(2) between 1 minute and 2 minute and

(3) 4 minute or less.

(AUC May / Jun 2012)


12

UNIT – IV: CORRELATION AND SPECTRAL DENSITIES:

PART -A (2 Marks)


4( ) 25

1 6XXR T

.

Find the mean and variance of the process. (AUC Nov/Dec 2011)

2. Prove that for a WSS process ( )X t , ( , )XXR t t is an even function of T.

(AUC Nov/Dec 2011)

(AUC Apr/May 2011)

3. Find the variance of the stationary process ( )X t whose auto correlation function is

given by 2

( ) 2 4XXR e


4. State any two properties of cross correlation function. (AUC Nov/Dec 2010)


9( ) 16

1 6R

.

Find the mean and variance of the process. (AUC Apr/May 2010)

(AUC Apr/May 2011)

(AUC May / Jun 2012)

6. Find the power spectral density function of the stationary process whose auto

correlation function is given by e

. (AUC Apr/May 2010)

7. Prove that ( ) ( ).xy yxS S (AUC May / Jun 2012)


1. The auto correlation function of a random process is given by

2

2

;

( ) 1( ) ;

T

R T TT

.Find the power spectral density of the process.

(AUC Nov/Dec 2011)

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

2

4

9( ) .

5 4xxS

Find the mean square value of the process. (AUC Nov/Dec 2011)

3. State and prove Weiner-Khintchine theorem. (AUC Nov/Dec 2011)

(AUC Nov/Dec 2010)

(AUC Apr/May 2011)

4. The cross- power spectrum of real random processes ( )X t and ( )Y t is given by

1( )

0 ,xy

a b forS

elsewhere

jω Find the cross correlation function.

(AUC Nov/Dec 2011)


13

(AUC Nov/Dec 2010)

(AUC Apr/May 2011)

5. If ( )X t and ( )Y t are two random processes with auto correlation function

( ) (0) (0)XX XX YYR T R R . Establish any two properties of auto correlation function

( )XXR T . (AUC Nov/Dec 2010)

(AUC Nov/Dec 2012)

6. Find the power spectral density of the random process whose auto correlation

function is ( )

1 ,

0 ,

for TR T

for T


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

0

( )

1 ,

0 ,S

otherwise

. Find 𝑅(𝑇) and show that 𝑋(𝑡) and

0

( )X t

are uncorrelated.

(AUC Apr/May 2011)

8. The auto correlation function of a WSS process is given by 22( )R e

determine the power spectral density of the process. (AUC Apr/May 2011)

9. Find the auto correlation function of the periodic time function ( ) sinX t t .

(AUC Apr/May 2010)

10. Find the autocorrelation function of the process ( )X t for which the power spectral

density is given 2( ) 1 1 ( ) 0 1XX XXS for and S for .

(AUC Apr/May 2010)

11. If ( )X t and ( )Y t are zero mean and stochastically independent random

processes having auto correlation function ( ) ( ) cos2XX YYR e and R t

respectively. Find (1) The autocorrelation function of

( ) ( ) ( ) ( ) ( ) ( )W t X t Y t and Z t X t Y t

(2) The cross correlation function of W(t) and Z(t).

(AUC Apr/May 2010)

12. A stationary random process X(t) with mean 2 has the auto correlation function

10( ) 4XXR e

. Find the mean and variance of

1

0

( ) .Y X t dt

(AUC May/Jun 2012)

13. Find the power spectral density function whose autocorrelation function is given by2

0( ) cos( )2

XX

AR . (AUC May/Jun 2012)

14. The cross-correlation function of two processes 𝑋(𝑡) and 𝑌(𝑡) is given by

0 0( , ) sin( ) cos 22

XX

ABR t t t where A, B and 0 are constants.

Find the cross-power spectrum SXY( ). (AUC May/Jun 2012)

15. Find the power spectral density of the random process whose auto correlation

function is 1 , 1

( )0 ,

forR

elsewhere

(AUC Nov/Dec 2012)


14

16. State and prove Wiener Khintchine theorem and hence find the power spectral

density of a WSS process X(t) which has an autocorrelation

0( ) [1 ],XXR A T T t T (AUC Nov/Dec 2012)


15

UNIT – V: LINEAR SYSTEMS WITH RANDOM INPUTS:

PART -A (2 Marks)

1. State any two properties of linear time-invariant system. (AUC Nov/Dec 2011)

2. If ( )X t and ( )Y t in the system ( ) ( ) ( )Y t h u X t u du

are WSS process, how

are their auto correlation functions related. (AUC Nov/Dec 2011)

3. If ( )Y t is the output of an linear time invariant system with impulse response ℎ(𝑡),

then find the cross correlation of the input function 𝑋(𝑡) and output function 𝑌(𝑡).

(AUC Nov/Dec 2010)

4. Define Band-Limited white noise. (AUC Nov/Dec 2011)

(AUC Apr/May 2011)

5. Find the system transfer function if a linear time Invariant system has an impulse

function. (AUC Apr/May 2011)

6. Define time-invariant system. (AUC Apr/May 2010)

7. State auto correlation function of the White noise. (AUC Apr/May 2010)

8. Prove that the system ( ) ( ) ( )Y t h u X t u du

is a linear time invariant system.

(AUC May/Jun 2012)

9. What is unit impulse response of a system? Why is it called so?

(AUC May/Jun 2012)


1. If the input to a time variant, stable, linear system is a WSS process, prove that the

output will also be a WSS process. (AUC Nov/Dec 2011)

2. Let X(t) be a WSS process which is the input to a linear time invariant system with

unit impulse h(t) and output Y(t), then prove that 2

( ) ( ) ( )yy xxS H S .

(AUC Nov/Dec 2011)

3. For a input-output linear system (𝑋(𝑡), ℎ(𝑡), 𝑌(𝑡)), derive the cross correlation

function RXY(T) and the output auto correlation function RYY(T).

(AUC Nov/Dec 2011)

4. Show that the input ( )X t is a WSS process for a linear system then output ( )Y t is

a WSS process. Also find RYY(T). (AUC Nov/Dec 2010)

5. If X(t) is the input voltage to a circuit and y(t) is the output voltage. ( )X t is a

stationary random process with μx =0 and 2

( )XXR e

. Find the mean μy and


16

power spectrum ( )yyS of the output if the system transfer function is given by

1( )

2H

i


6. If 0( ) cos( ) ( ),Y t A t N t where A is a constant, θ is a random variable with a

uniform distribution in (-π, π) and ( )N t is a band-limited Gaussian White noise with

power spectral density 0

0,( ) 2

0 ,

BNN

Nfor

S

elsewhere

find the power spectral

density ( )Y t . Assume that ( )N t and θ are independent.

(AUC Nov/Dec 2010)

(AUC Apr/May 2010)

7. A system has an impulse response ( ) ( )h t e U t t. Find the power spectral density

of the output 𝑌(𝑡), corresponding to the input 𝑋(𝑡). (AUC Nov/Dec 2010)

8. Consider a system with transfer function 1

.1 j

An input signal with auto correlation

function 2( )m m is fed as input to the system. Find the mean and mean square

value of the output. (AUC Apr/May 2011)

(AUC May/Jun 2012)

9. A stationary random process X(t) having the autocorrelation function ( ) ( )XXR A

is applied to a linear system at time t=0 where ( )f represent the impulse function.

The linear system has the impulse response of ( ) ( ) ( )h t e u t u t btwhere represents

the unit step function. Find ( ).YYR Also find the mean and variance of 𝑌(𝑡).

(AUC Apr/May 2011)

(AUC May/Jun 2012)

10. A linear system is described by the impulse response 1

( ) ( ).h t e u tRC

-tRC Assume an

input process whose Auto correlation function is ( )B . Find the mean and auto

correlation function of the output process. (AUC Apr/May 2011)

11. If ( )N t is a band limited white noise centered at a carrier frequency ω0 such that

00,

( ) 2

0 ,

BNN

Nfor

S

otherwise

. Find the auto correlation of ( )N t .

(AUC Apr/May 2011)

(AUC May/Jun 2012)


17

12. A wide sense stationary random process ( )X t with auto correlation ( )XXR e

where A and ‘a’ are real positive constants, is applied to the input of an Linear

transmission input system with impulse response ( ) ( )h t e u t bt where b is a real

positive constant. Find the auto correlation of the output 𝑌(𝑡) of the system.

(AUC Apr/May 2010)

13. If X(t) is a band limited process such that

2 2( ) 0 , 2 (0) ( ) (0)XX XX XX XXS R R R when provethat

(AUC Apr/May 2010)

14. Assume a random process X(t) is given as input to a system with transfer function

0 0( ) 1 .H for If the auto correlation function of the input process is

0 ( ).2

N Find the auto correlation function of the output process.

(AUC Apr/May 2010)

Probability Random Process QB Mahalakshmi Engg

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