ma6451- probability and random process question …

CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY

(Approved by AICTE New Delhi, Affiliated to Anna University Chennai.

Zamin Endathur Village, Madurntakam Taluk, Kancheepuram Dist.-603311.)

MA6451- PROBABILITY AND RANDOM PROCESS

QUESTION BANK

(YEAR/SEM: II/IV)

UNIT – I

RANDOM VARIABLES

PART – A (2 Marks)

1. X and Y are independent random variables with variance 2and 3. Find the variance of .

(Apr/May-2014)

2. A continuous random variable X has probability density function (pdf)

find k such that (Apr/May-2014)

3. A random variable X has cdf Find the pdf of X and the expected

value of X. (Apr/May-2013)

4. Find the moment generating function of binomial distribution. (Apr/May-2013)

5. Find C, if (Apr/May-2012)

6. The probability that a man shooting a target is ¼. How many times must he fire so that the

probability of his hitting the target atleast once is more than 2/3? (Apr/May-2012)

7. Find CDF of a continuous random variable is given by .Find the PDF

and mean of X. (Apr/May-2011/Nov-2011)

8. Establish the memoryless property of the exponential distribution. (Apr/May-2011)

9. If the PDF of a random variable X is find .

(Apr/May-2010)

10. If the MGF of a uniform distribution for a random variable X is , find E(X).

(Apr/May-2010)

11. Find c, if a continuous random variable X has the density function .

(Nov/Dec-2014) 12. Find the moment generating function of Poisson distribution. (Nov/Dec-2014)

13. Define Random Variable (Nov/Dec-2013)

14. Define geometric distribution. (Nov/Dec-2013).

15. If X is a normal random variable with mean zero and variance find the PDF of .

(Nov/Dec-2011)

16. A continuous random variable X has probability density function

Find k such that . (Nov/Dec-2010)

www.vidyarthiplus.com


17. If X is uniformly distributed in . Find the pdf of (Nov/Dec-2010).

18. Define discrete random variable with an example.

19. Define Poisson distribution

20. For a binomial distribution mean is 6 and S.D is . Find the first two terms of the distribution.

21. Define Exponential distribution.

22. Define Gamma distribution.

23. Define Normal distribution.

24. State the properties of moment Generating function.

25. If X is uniformly distributed with mean 1 and variance 4/3, find .

PART-B (16 Marks) 1. Define the moment generating function (MGF) of a random variable?. Derive the MGF, mean

variance and the first four moments of a Gamma distribution. (Apr/May-2014)

2. Describe Binomial B distribution and obtain the moment generating function. Hence compute

(1) the first four moments and (2) the recursion relation for the central moments.

(Apr/May-2014/2012) 3. A random variable X has the following probability distribution

X 0 1 2 3 4 5 6 7

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

Find (1) The value of K.

(2) and

(3) The smallest value of n for which . (Apr/May-2014/Nov-2010)

4. Find the MGF of a random variable X having the PDF .Also deduce

first four moments about the origin. (Apr/May-2014/2012)

5. A random variable X has pdf . Find the rth moment of X about origin.

Hence find the mean and variance. (Apr/May-2013)

6. A random variable X is uniformly distributed over (0,10). Find

(1) and (2) (Apr/May-2013)

7. An office has four phone lines. Each is busy about 10% of the time Assume that the phone lines act

independently.

(1) What is the probability that all four phones are busy?

(2) What is the probability that atleast two of them are busy? (Apr/May-2013)

8. Describe gamma distribution. Obtain its moment generating function. Hence compute its mean and

variance. (Apr/May-2013).

9. Given that X is distributed normally, if and , find the mean

and standard deviation of the distribution. (Apr/May-2012).

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

. (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? (Apr/May-2012).



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

(1) Find the value of ‘k’ (2) Find (3) What is

(4) Find the distribution function of (Apr/May-2011).

12. Derive the M.G.F of poisson distribution and hence or otherwise deduce its mean and variance.

(Apr/May-2011). 13. 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

set, what is the probability that exactly 2 of them will have marks over 70? (Apr/May-2011).

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

, where and if

Find (1) the value of C (2) . (3) The distribution function of X.

(4) The largest value of X for which . (Apr/May-2010).

15. 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 (1) on the fourth trial and (2) in less

than 4 trials? (Apr/May-2010).

16. Find the MGF of the two parameter exponential distribution whose density function is given by

and hence find the mean and variance. (Apr/May-2010).

17. 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 two of them will have marks over 70? (Apr/May-2010).

18. Find the nth moment about mean of normal distribution. (Nov/Dec-2014)

19. Derive Poisson distribution from the binomial distribution. (Nov/Dec-2014/2013)

20. Find the mean and variance of Gamma distribution. (Nov/Dec-2014/2013)

21. A random variable X has the pdf . Obtain the mgf and first four moments

about the origin. Find mean and variance of the same. (Nov/Dec-2014)

22. Suppose that a customer’s arrive at a bank according to a Poisson process with mean rate of 3 per

minute. Find the probability that during a time interval of 2min. (1) exactly 4 customers arrive

and (2) more than 4 customers arrive. (Nov/Dec-2013)

23. If X and Y are independent RVs each normally distributed with mean zero and variance , find

the pdf of and . (Nov/Dec-2013)

24. Find the moment generating function, mean and variance for the random variable X having the

probability density function (Nov/Dec-2012).

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

Find (1) Mean of X, (2) (3) P[X is divisible by 3]. (Nov/Dec-2011).



UNIT – II

TWO DIMENSIONAL RANDOM VARIABLES

PART – A (2 Marks) 1. State the Central limit theorem for II random variable. (May/June-2014/May-2012/Nov/Dec 2010).

2. State the basic properties of joint distribution of (X, Y) when X and Y are random variables.

(M/J-14) 3. The joint pmf of two random variables X and Y is given by

Determine the value of the constant k. (May/June-2013)

4. The joint pdf of a random variable (X,Y) is

FindP{X<Y}. (May/June-2013).

5. Let X and Y be two discrete random variables with joint probability mass function

Find the marginal probability mass functions of X and Y. (May/June – 2012)

6. Let X and Y be continuous random variables with joint probability density function

Elsewhere. Find (y/x).

(May 11)&(Nov/Dec – 2010)

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

(May11) 8. Find the value of k, if f (x, y) = k (1- x)(1- y) in 0 < x, y < 1and f (x, y) = 0 , otherwise, is to be the

joint density function. (May/June – 2010)

9. A random variable X has mean 10 and variance 16. Find the lower bound for P(5 < X < 15) .

(May/June – 2010)

10. Given the random variable X with density function f(x) = . Find the pdf of Y=8X3

(Nov/Dec -2014)&(Nov/Dec – 2013)

11. Define the joint pmf of a two dimensional discrete random variables.(Nov/Dec-2014)

12. Define Random Process? (Nov/Dec – 2013)

13. State and prove memory less property of exponential distribution.(Nov/Dec – 2012)

14. Define Chebyshev inequality. (Nov/Dec-2012)

15. If the joint pdf of (X, Y) is Check whether X and Y are I

independent. (Nov/Dec-2012)

16. The regression equations are 3x+2y=26 and 6x+y = 31. Find the correlation coefficient between X

and Y. (Nov/Dec-2012)

17. Define Two dimensional random variable.

18. Give any two properties of correlation coefficient.

19. Distinguish between correlation and regression.



20. If joint pdf of the random variables is given by . Find

the value of k.

21.Show that .

22. If X and Y are independent random variables with variance 2 and 3. Find the variance of 3X+4Y.

23.If X and Y have the joint density function . Find .

24. Find the covariance of the two RV’s whose pdf is given by

25.Write a note aon Central limit theorems.

PART-B (16 Marks)

1. If the joint pdf of a two dimensional random variable is given by

. Find (1) (2) (3) and

(4) Find the conditional density functions. (Apr/May-2014)

2. The joint p.d.f of the random variable is

, find Cov (X,Y) (Apr/May-2014)

3. Marks obtained by 10 students in Mathematics (x) and Statistics (y) are given below:

x: 60 34 40 50 45 40 22 43 42 64

y: 75 32 33 40 45 33 12 30 34 51

Find the two regression lines. Also find y when . (Apr/May-2014)

4. Two Independent random variables X and Y are defined by

and Show that U = X+Y and V = X-Y are

uncorrelated. (May/June-2013)

5. State and prove the central limit theorem for in the case of iid random variables. (May/June-2013)

6. The equations of two regression lines are 3x+12y = 19 and 3y+9x = 46. Find and the

correlation Coefficient between X and Y. (May/June-2013)

7. Given the joint pdf of X and Y

(1) Evaluate C.

(2) Find marginal pdf of X.

(3) Find the conditional density of Y|X. (May/June-2013)

8. The joint probability density function of the random variable(X,Y) is given by

f(x,y) = Kxy , x>0, y>0. Find the value of K and Cov(X, Y). are X and Y

independent?(May/June-2012)

9. If X and Y are uncorrelated random variables with variances 16 and 9. Find the correlation co

efficient between X+Y and X-Y. (May/June-2012)

10. Let(X,Y) be a two dimensional random variable and the probability density function be given by

f(x,y) = x+y, 0 x, y . Find the pdf of U = XY. (May/June-2012)

11. Let be Poisson variates with parameter and

where n =75. Find P [120 using central limit theorem. (May/June-2012)



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

Calculate the conditional distribution of X, given that X +Y = n. (May/June-2011)

13. The regression equation of X on Y is 3Y - 5X + 108 = 0 .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. (May/June-2011)

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

, find the density function of Z=XY.(May/June-2011)

15. The life time of a particular variety of electric bulb may be considered as a random variable with

mean 1200 hours and standard deviation 250 hours. Using central limit theorem, find the

probability that the average life time of 60 bulbs exceeds 1250 hours. (May/June-2011)

16. Find 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. (May/June-2010)

17. Find the covariance of X and Y, if the random variable (X, Y) has the joint p.d.f

f (x, y) = x + y, 0 x 1, 0 y 1and f (x, y) = 0 , otherwise. (May/June-2010)

18. 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. (May/June-2010)

19. The joint p.d.f of two dimensional random variable (X, Y) is given by f (x, y) = xy , 0 x y 2

and f (x, y) = 0 ,otherwise. Find the densities of X and Y, and the conditional densities f (x / y) and

f (y / x). (May/June-2010)

20. State and prove the central limit theorem. (Nov/Dec-2014).

21. 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 central limit theorem that the average lifetime

of 60bulbs exceed 1250h. (Nov/Dec-2014).

22. 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 K and all the marginal and conditional probability distributions. Also find the probability

distribution of (X+Y). (Nov/Dec-2014).



23. Find the moment generating function of a binominal distribution and find mean and variance.

(Nov/Dec – 2012)

24. State and prove the additive property and Poisson distribution. (Nov/Dec – 2012)

25. If are uniform variables with mean = 2.5 and variance = ¾, use CLT to estimate

P (108 where , n =48. (Nov/Dec – 2011)

UNIT – III

RANDOM PROCESSES

PART – A (2 Marks) 1. State the properties of an ergodic process. (Apr/May-2014)

2. Explain any two applications of binomial process. (Apr/May-2014)

3. Define wide sense stationary process. (Apr/May-2013/2012/2010)

4. Show that a binomial process is Markov. (Apr/May-2013)

5. If is a normal process with and find the variance of

X(10)-X(6). (Apr/May-2012/Nov-2011)

6. Prove that a first order stationary process has a constant mean. (Apr/May-2011)

7. State the postulates of a Poisson process. (Apr/May-2011/(Nov/Dec-2010).

8. Define a Markov chain and give an example. (Apr/May-2010)

9. Define stochastic processes. (Nov/Dec-2014)

10. Define Markov processes. (Nov/Dec-2014/2013)

11. Define random process.(Nov/Dec-2013)

12. What is a random process said to be mean ergodic? .(Nov/Dec-2011)

13. Find for the joint probability density function

. .(Nov/Dec-2012)

14. Define Liapounoff’s form of central limit theorem. .(Nov/Dec-2012)

15. Consider the random process where is a random variable with density

function . Check whether or not the process is wide sense stationary. .

(Nov/Dec-2010)

16. Give an example of Markov process.

17. Give an example of continuous time random process.

18. Define a Stationary process.

19. State the four types of a stochastic processes.

20. Define ergodic random process.

21. Give an example of an Ergodic process.

22. Define irreducible Markov chain?

23. Define transition probability matrix.

24. State any two properties of a Poisson process.

25. Define Poisson random process.



PART-B (16 Marks)

1. The process whose probability distribution under certain condition is given by

Find the mean and variance of the process, Is the process first-

order stationary? (Apr/May-2014/Nov-2014/2011)

2. If the WSS process is given by , where is uniformly distributed

over prove that is correlation ergodic. (Apr/May-2014/2012)

3. If the Process is aPoisson process with parameter , obtain ; Is the

process first order stationary? (Apr/May-2014)

4. Prove that a random telegraph signal process is a Wide Sense Stationary Process

Where is a random variable which is independent of and assumes values and +1 with

equal probability and . (Apr/May-2014)

5. Define a semi random telegraph signal process and prove that it is evolutionary. (Apr/May-2013)

6. Mention any three properties each of auto correlation and of cross correlation functions of a wide

sense stationary process. (Apr/May-2013/Nov-2014)

7. A random process defined by where A and B are

independent random variables each of which has a value -2 with probability and a value 1 with

probability 2/3. Show that is a wide sense stationary process. (Apr/May-2013/2011)

8. Define a Poisson process. Show that the sum of two Poisson processes is a Poisson process.

(Apr/May-2013)

9. If is a WSS process with autocorrelation determine the second order moment

of the RV . (Apr/May-2012)

10. If 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 !

minute, (2) between 1 minute and 2 minute and (3) 4min or Less. (Apr/May-2012)

11. Suppose that is a Gaussian process with ,

Find the probability that . (Apr/May-2012)

12. A random process has sample functions of the form , where is

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

that is uniformly distributed between 0 and . Assume that the random variables A and are

independent. Is is a mean – ergodic process? (Apr/May-2011)

13. If is a Gaussian process with and find the probability

that (1) . (2) . (Apr/May-2011/Nov-2014/2013)

14. Prove that the interval between two successive occurrences of a Poisson process with parameter

has an exponential distribution with mean (Apr/May-2011)

15. Examine whether the random process is a wide sense stationary if A

and are constants and is uniformly distributed random variable in .

(Apr/May-2010/Nov-2011)



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

duration t seconds, is a Poisson process with mean . (1) The probability that exactly 3

messages will arrive during 10 second interval (2) The probability that the number of message

arrivals in an interval of duration 5 seconds is between 3 and 7. (Apr/May-2010)

17. The random binary transmission process is a wide sense process with zero mean and

autocorrelation function , where T is a constant. Find the mean and variance of the

time average of over . Is mean – ergodic? (Apr/May-2010)

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

is , and the initial distribution is , Find

and . (Apr/May-2010)

19. If the 2n random variables and are uncorrelated with zero mean and

, show that the process, is wide

sense stationary. What are the mean and autocorrelation of ? (Nov/Dec-2014/2013)

20. Define Random telegraph signal process and prove that it is wide-sense stationary.

(Nov/Dec-2013)

21. Prove that sum of two independent Poisson processes is a Poisson process. (Nov/Dec-2013).

22. The cumulative distribution function of two discrete random variables X and Y is given as

follows: Determine the (1) Joint probability mass function of X and Y.

(2) Marginal probability mass function of X and Y. (Nov/Dec-2012)

23. The joint probability density function of X and Y is given by .

Find the probability density function . (Nov/Dec-2012)

24. Find the correlation co-efficient and also the conditional density functions for the joint

probability density function (Nov/Dec-2011)

25. 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. (Nov/Dec-2011)

UNIT IV

CORRELATION AND SPECTRAL DENSITIES

PART- A (2 Marks) 1. Define Cross correlation function and state any two of its properties.(Apr/May-2014)

2. Define a System. When is it called a linear system. (Apr/May-2014)

3. A random process is deifined by where is a constant and K is

uniformly distributed over . Find the auto correlation function of (Apr/May-2013)

4. Define cross correlation function of and . When do you say that they are independent?

(Apr/May-2013)



5. The autocorrelation function of a stationary random process is , Find the mean

and variance of the process. (Apr/May-2012/2011/2010)

6. Prove that (Apr/May-2012)

7. Prove that for a WSS process , is an even function of . (Apr/May-2011) 8. Find the power spectral density function of the stationary process whose autocorrelation function is

given by . (Apr/May-2010) 9. Define power spectral density function. (Nov-2013)

10. State Wiener –Khinchine theorem. (Nov-2013)

11. Find the variance of the stationary process whose auto correlation function is given by

(Nov-2012/2010) 12. Prove that for a WSS process is an even function of . (Nov-2012/2011)

13. The autocorrelation function of a stationary random process is , Find the mean

and variance of the process. (Nov-2011) 14. Find the mean of the stationary process , whose autocorrelation function is given by

. (Apr/May-/2011)

15. State any two properties of cross correlation function. (Nov-2010).

16. Define autocorrelation function and prove that for a WSS process .

17. State any two properties of auto correlation function

18. Define cross – spectral density

19. Define spectral density.

20. What is mean by spectral analysis?

21. Give an example of Cross spectral density.

22.If is the auto correlation function of a random process obtain the spectral

density of X(t).

23. The power spectral density of a random process is given by

Find its autocorrelation function.

PART - B(16 Marks) 1. Find the mean and auto correlation of the Poisson process.(Apr/May-2014)

2. Prove that the random processes and defined by and

are jointly side sense stationary.(Apr/May-2014)

3. State and Prove Weiner- Khintchine Theorem (Apr/May-2014/2013/2011/Nov-2013/2011)

4. Define spectral density of a stationary random process . Prove that for a real random process

the power density is an even function. (Apr/May-2013)

5. Two random processes and are defined as follows: and where A, B and are constants; is a uniform random variable over

, Find the cross correlation function of and .(Apr/May-2013) 6. If the cross power spectral density of and is

where a and b are constants. Find the cross

correlation function.(Apr/May-2013)



7. A stationary random process with mean 2 has the auto correlation function

find the mean and variance of (Apr/May-2012)

8. Find the power spectral density function whose autocorrelation function is given by

. (Apr/May-2012)

9. The cross correlation function of two processes and is given by

where A, B and are constants. Find the cross- power

spectrum . (Apr/May-2012) 10. Let and be both zero mean and WSS random processes consider the random process

defined by + . Find (i) The Auto correlation function and the power

spectrum of if and are jointly WSS. (ii) The power spectrum of Z(t) if and

are orthogonal. (Apr/May-2012) 11. The cross –power spectrum of real random processes and is given by

Find the cross correlation function.

(Apr/May-2011/Nov-2013) 12. The power spectral density function of a zero mean WSS process is given by

. Find and show that and are uncorrelated.

(Apr/May-2011) 13. The Auto correlation function of a WSS process is given by determine the

power spectral density of the process.(Apr/May-2011) 14. Find the autocorrelation function of the periodic time function of the period time function

. (Apr/May-2010) 15. The autocorrelation function of the random binary transmission is given by

for and for . Find the power spectrum of the

process . (Apr/May-2010) 16. and are zero mean and stochastically independent random processes havind

autocorrelation functions and respectively. Find (i) The

autocorrelation function of and (ii) The cross correlation

function of W(t) and Z(t). (Apr/May-2010) 17. Find the autocorrelation function of the process for which the power spectral density is

given by for and for .(Apr/May-2010) 18. The autocorrelation function of the random twlegraph signal process is given by

determine the power density spectrum of the random telegraph signal.

(Nov-2013) 19. The autocorrelation function of the Poisson increment process is given by (Nov-2013)

prove thati its spectral density is

.



20. If the power spectral density of a WSS process is given by

Find the autocorrelation function of the process. (Nov-2013)

21. If the process is defined as where and ) are independent

WSS process es, prove that(1) and

(2) . (Nov-2013)

22. If and are two random processes with auto correlation function

respectively the n prove that . Establish any two properties of auto

corrleation function . (Nov-2012)

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

. (Nov-2012)

UNIT – V

LINEAR SYSTEMS WITH RANDOM INPUTS

PART – A(2 Marks)

1. Define a system. When is it called a linear system? (Apr/May-2014)

2. Define Band –Limited white noise. (Apr/May-2014/ Nov-2012/2011)

3. Define a linear time invariant system. (Apr/May-2013)

4. State the convolution form of the output of a linear time invariant system. (Apr/May-2013)

5. Prove that the system is a linear time-invariant system.

(Apr/May-2012) 6. What is unit impulse response of a system? Why is it called so? (Apr/May-2012)

7. Find the system Transfer function, if a Linear Time Invariant system has an impulse function

(Apr/May-2011/ Nov-2012)

8. Define White noise. (Apr/May-2011/Nov-2013)

9. Define Time-Invariant system. (Apr/May-2010 Nov-2013)

10. State autocorrelation function of the white noise. (Apr/May-2010)

11. State any two properties of a linear time invariant system. (Nov-2011)

12. If and in the system are WSS process how are their

autocorrleation functions related (Nov-2011)

14. If is the output of an linear time invariant system with impulse response . Then find the

cross correlation of the input function and output function . (Nov-2010)

14. Describe a linear system.

15. State the properties of a linear filter.

PART – B (16 Marks)

1. Show that if the input is a WSS process for a linear system then output is a WSS

process. Also find . (Apr/May-2014/Nov-2012)

2. If is the input voltage to a circuit and is the output voltage , is a stationary

random process with and Find the mean and power spectrum

of the out if the power transfer function is given by (Apr/May-2014)



3. If , where A is a constant , is a random variable with a uniform

distribution in and is a band limited Gaussian white noise with power spectral

density Find the power spectral density . Assume that

and are independent. (Apr/May-2014)

4. A System has in impulse response , find the power spectral density of the output

corresponding to the input . (Apr/May-2014/Nov-2012)

5. A random process is the input to a linear system whose impulse function is

The auto correlation function of the process is and Find the power

spectral density of the output process . (Apr/May-2013)

6. A side sense stationary noise process has an auto correlation function

where P is a constant. Find its power spectrum. (Apr/May-2013)

7. If the input to a time invariant stable, linear system is a wide sense stationary process, prove that

the output will also be a wide sense stationary process.(Apr/May-2013)

8. Let be a Wide sense stationary process which is the input to a linear time invariant system

with unit impulse and output , then prove that where

is Fourier transform of . (Apr/May-2013)

9. Consider a system with transfer function an input signal with autocorrelation function

is fed as input to the system. Find the mean and mean-square value of the output.

(Apr/May-2012/2011) 10. A stationary random process having the autocorrelation function and is

applied to a liear system at time t=0 where represent the impulse fuction. The linear system

has the impulse response of where represents the unit step function. Find

. Also find the mean and variance of . (Apr/May-2012/2011)

11. If is a WSS process and if then prove that (i)

where * stands for convolution. (ii) . (Apr/May-2012)

12. If is a band limited white noise centered at a carrier frequency such that

Find the autocorrelation of . (Apr/May-2012/2011)

13. A linear system is decribed by the impulse response . Assume an input

process whose Auto correlation function is . Find the mean and Auto correlation function of

the output process. (Apr/May-2011)

14. A wide sense stationary random process with autocorrelation where A and

a are real positve constants, is applied to the input of an Linear transmission input system with

impulse response where b is a real positive constant. Find the autocorrelation of

the output of the system. (Apr/May-2010)

15. If is aband limited process such that when , prove that

. (Apr/May-2010)



16. Assume a random process is given as input to a system with transfer function for

If the autocorrelation function of the input process is , find the

autocorrelation function of the output process. (Apr/May-2010)

17. If where A is a constant , is a random variable with a uniform

distribution in and is a band limited Gaussian white noise with power spectral

density Find the power spectral density . Assume that

and are independent. (Apr/May-2010- Nov-2013/2012)

18. Prove that the spectral density of any WSS process is non-negative. (Nov-2013)


random process with and Find the mean , and if the

power transfer function is given by (Nov-2013)


random process with and Find the mean and power spectrum

of the out if the power transfer function is given by (Nov-2012)

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