probability random process qb 2

181403 - Probability & Random Process

Department of Electronics and Communication Engineering

Question BankSUBJECT CODE : 181403 SUBJECT NAME : PROBABILITY & RANDOM PROCESS BRANCH / YEAR : ECE / II

UNIT –I

(RANDOM VARIABLES) Part – A

1. Define random variable.2. X and Y are independent random variables with variances 2 and 3. Find the variance of 3X +4Y.

4. The number hardware failures of a computer system in a week of operations as the following pmf:Number of failures: 0 1 2 3 4 5 6 Probability : 0.18 0.28 0.25 0.18 0.06 0.04 0.01

Find the mean of the number of failures in a week.5. A continuous random variable X has the probability density function given by 10,3)( 2 ≤≤= xxxf . Find

K such that P(X > K)= 0.5

6. A random variable X has the pdf f(x) given by

≤>

=−

0,00,

)(x

xCxexf

x

. Find the value of C and c.d.f of X.

7. The cumulative distribution function of a random variable X is F(x)= [ ] 0,)1(1 >+− − xex x . Find the probability density function of X.

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

9.Let X be a R.V with p.d.f given by <<

=otherwise

xxxf

,010,2

)( . Find the pdf of Y =(3X +1).

10.Find the cdf of a RV is given by

<

≤≤

<

=

x

xxx

xF

4,1

40,16

0,0

)(2

and find P(X>1/X<3).

11.A continuous random variable X that can assume any value between x = 2 and x = 5 has a density function given by f(x) = K(1 + x). Find P[X<4]. 12.The first four moments of a distribution about x = 4 are 1, 4, 10 and 45 respectively. Show that the mean is 5, variance is 3, 03 =µ and .264 =µ 13. Define moment generating function. 14.Write down the properties of moment generating function.

3. Let X be a R.V with E[X]=1 and E[X(X-1)]=4 . Find var X and Var(2-3X).


15.Find the moment generating function for the distribution where

=

=

=

otherwise

x

x

xf

,0

2,31

1,32

)( .

16. For a binomial distribution mean is 6 and S.D is 2 . Find the first two terms of the distribution. 17. Comment the following: the mean of a binomial distribution is 3 and variance is 4. 18. Find the moment generating function of binomial distribution. 19.The mean of a binomial distribution is 20 and standard deviation is 4. Find the parameters of the distribution. 20.If X is a Poisson variate such that P(X=2) = 9P(X=4) +90P(X=6),find the variance. 21. Define Poisson distribution and state any two instances where Poisson distribution may be successfully employed. 22. The number of monthly breakdown of a computer is a random variable having a Poisson distribution with mean equal to 1.8. Find the probability that this computer will function for a month . (i) Without a breakdown (ii) With only one breakdown (iii) With at least one breakdown. 23. In which probability distribution , variance and mean are equal. 24. One percent of jobs arriving at a computer system need to wait until weekends for scheduling , owing to core-size limitations. Find the probability that among a sample of 200 jobs there are no job that have to wait until weekends. 25. Write the MGF of geometric distribution. 26. If the probability is 0.10 that a certain kind of measuring devic will show excessive drift, what is the probability that the fifth measuring device tested will be the first show excessive drift? Find its expected value also.

27. If X is uniformly distributed in

−

2,

2ππ

, find the probability distribution function of y= tan x.

28.Show that for the uniform distribution axaa

xf <<−= ,21)( the moment generating function about origin

is at

atsinh.

29.Let X be a uniform random variable over [-1,1].Find (a) )43())(3

1( ≥< XPbXP . 30.A fast food chain finds that the average time customers have to wait for service is 45 seconds. If the waiting time can be treated as an exponential random variable, what is the probability that a customer will have to wait more than 5 minutes given that already he waited for 2 minutes? 31.Define Generalised form of the gamma distribution . 32. Write two characteristic of the normal distribution. 33.If X is a Gaussian random variable with mean zero and variance 2σ , find the probability density function of XY = .

34.A random variable X has p.d.f

<>

=−

0,00,

)(xxe

xfx

. Find the density function of x1

A.R.Engineering College / Villupuram 2


35. State Memoryless property of exponential distribution.

Part- B 1.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 k2 2k2 7k2+k

(i)Find k.

(ii) Evaluate P(X<6),P(X 6≥ ).

(iii)If P(X C≤ )>1/2, find the maximum value of C.

(iv)Evaluate P(1.5<X<4.5/X>2).

(v)Find P(X<2),P(X>3),P(1<X<5). (8)

2.If X has the distribution function

≥

<≤

<≤

<≤<

=

10,1

106,65

64,21

41,31

1,0)(

x

x

x

xx

XF . Find

(i)The probability distribution of X.

(ii)P(2<X<6).

(iii)mean of X.

(iv)variance of X. (8)

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

≤≤−<<

=otherwise

xxkxx

xf X

,021),2(

10,)(

.

(i)Find the value of k.

(ii)Find P(0.2<X<1.2)


(iii)What is P(0.5<X<1.5/X 1≥ )

(iv)Find the distribution function of f(x). (8)

4. Find the moment generating function of a random variable X having the probability density function

>=

−

otherwise

xexfx

,0

0,21

)(2

and hence find the mean and variance. (8)

5. A continuous random variable X has the p.d.f f(x)= K 0,2 ≥− xex x , Find the rth moment

of X about the origin. Hence find mean and variance of X. (8)

6.Find the probability distribution of the total number of heads obtained in four tosses of a balanced coin.

Hence obtain the MGF of X , mean of X and variance of X. (8)

7. Derive mean and variance of binomial distribution. (8)

8.Out of 800 families with 4 children each, how many families would be expected to have (i)2 boys and 2 girls

(ii) atleast 1 boy (iii) atmost 2 girls and (iv) children of both genders. Assume equal probabilities for boys and

girls. (8)

9. Obtain the MGF of Poisson distribution and hence compute the first four moments. (8)

10. Derive MGF of Poisson distribution and hence find mean and variance. (8)

11.The sum of two independent poisson variates is a poisson variate. (8)

12.A manufacturer of television sets knows that of an average 5% of his product is defective. He sales

television in consignment of 100 and guarantees that not more than 4 sets will be defective . What is the

probability that a television set will fail to meet the guaranteed quality? (8)

13. Prove the memory less property of the Geometric distribution. (8)

14.If the probability that an applicant for a driver’s licence will pass the road test on any given trial is 0.8 ,

what is the probability that he will finally pass the test (a)on the fourth trial and (b)in fewer than 4 trails? (8)

15.X is uniformly distributed with mean 1 and variance 4/3 , find P(X<0). (8)

16.If X is a random variable uniformly distributed in (0 ,1), find the pdf of y= sinx . Also find the mean and

variance of y. (8)

17. Obtain the MGF of exponential distribution and hence compute the first four moments. (8)

18. The daily consumption of milk in excess of 20,000 gallons is approximately exponential with

θ = 3000.The city has a daily stock of 35,000 gallons. What is the probability that of two days selected at

random, the stock is insufficient for both days. (8)

19. Define Gamma distribution and find the mean and variance of the same. (8)


20. The daily consumption of milk in a city, in excess of 20,000 L is approximately

distributed as a Gamma variate with the parameters α =2 and =λ 000,101

.The city

has a daily stock of 30,000 litres. What is the probability that the stock is insufficient

on a particular day? (8)

21. State and explain the properties of Normal ),( 2σµN distribution . (8)

22.X is a normal variate with mean 30 and S.D 5. Find the probabilities that

(i) 530)(45)(4025 >−≥≤≤ XiiiXiiX .(8)

23.The marks obtained by a number of students for a certain subject is assumed to be normally distributed with

mean 65 and S.D of 5. If 3 students are taken at random from this set , what is the probability that exactly two

of them will have marks over 70? (8)

24.If X and Y are independent random variables each following N(0 ,2), find the probability density function

of z =2x +3y. (8)

25. Let X and Y be independent random variables with common p.d.f fX(x) = e-2x , x>0.

Find the joint pdf of U=X+Y and V=ex . (8)

UNIT –II

(TWO DIMENSIONAL RANDOM VARIBLES)

Part – A1.The joint pdf of two random variable X and Y is given by xyxxyxxxyf XY <<−<<−= ;20);(

81)( and

otherwise find

xyf

XY .

2. If two random variables X and Y have probability density function (PPF) )2(),( yxkeyxf +−= for x ,y >0,evaluate k.

3.Define joint probability distribution of two random variables X and Y and state its properties.4.If the point pdf of (X,Y) is given by 0,0),( )( ≥≥= +− yxeyxf yx find E[XY].

5.If X and Y have joint pdf <<<<+

=otherwise

yxyxyxf

,010,10,

),( , check whether X and Y are ndependent.

6. Find the marginal density functions of X and Y if 10,10),52(52),( ≤≤≤≤+= yxyxyxf .

7.If the function 1,10,10),1)(1(),( <<<<<−−= yyxyxcyxf to be a density function,find the value of c.


8.Let X and Y be continuous RVs with J.p.d.f

<<<<+=

otherwise

yxyxyyxf,0

10,10,232),(

2

.Find P(X +Y<1).

9.The regression lines between twqo random variables X and Y is given by 3X +Y =10 and 3X +4Y =12. Find the co-efficient of correlation between X and Y. 10.If X and Y are random variables such that Y = aX +b where a and b are real constants, show that the correlation co-efficient r(X,Y) between that has magnitude one. 11.If Y = -2X +3 , find the cov(X,Y).12.Let (X,Y) be a two dimensional random variable. Define covariance of (X,Y). If X and Y are independent , what will be the covariance of (X,Y).13. Prove that Cov(aX+bY)=abCov(X ,Y).14.Find the angle between the two lines of regression.15.The regression equations of X on Y and Y on X are respectively 5x –y =22 and 64x -45y =24. Find the means of X and Y.

16.The tamgent of the angle between the lines of regression Y on X and X on Y is 0.6 and yx σσ21= . Find the

correlation coefficient.17. Distinguish between correlation and regression.18.State the central limit theorem for independenc and identically distributed random variables.19.Write the applications of central limit theorem.

Part- B

1. The joint probability of 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) and P(X +Y>3). (8)

2.The two dimensional random variable (X,Y) has the joint density function

2,1,0;2,1,0,27

2),( ==+= yxyxyxf .Find the conditional distribution of Y given X = x . Also find the

conditional distribution of X given Y =1. (8)

3. The joint probability mass function (PMF) of X and Y is

Y 0 1 2

0 0.1 0.04 0.02

1 0.08 0.20 0.06


2 0.06 0.14 0.30

Compute the marginal PMF of X and of Y )1,1( ≤≤ YXP , and check if X and Y are independent. (8)

4. If the joint probability density function of a two dimensional random variable (X,Y) is given by

<<<<+=

elsewhere

yxxyxyxf,0

20,10,3),(

2

. Find

(i) P(X>1/2)

(ii) P(Y>k) and

(iii) P(Y<1/2/ X<1/2).

(iv)P(Y<1)

(v) Find the conditional density functions . (8)

5. Suppose the joint probability density function is given by

≤≤≤≤+=

otherwise

yxyxyxf,0

10,10),(56

),(2

.Obtain

the marginal PDF of X and that of Y.Hence or otherwise find )43

41( ≤≤ YP .(8)

6.X and Y are two random variables having joint density function

<<<<−−=

otherwise

yxyxyxf,0

42,20),6(81

),( .

Find (i) ( )31 <∩< YXP (or)P(X<1,Y<3) (ii)P(X+Y<3) (iii)P(X<1/Y<3). (8)

7. Two dimensional random variable (X,Y) has the joint PDF

f(x,y) =2,0<y<x<1; 0 otherwise. Find

(i) marginal and conditional distributions

(ii) joint distribution function F(x,y)

(iii) Check whether X and Y are independent.

(iv) P(X<1/2/Y<1/4) . (8)

8. Two dimensional random variable (X,Y) has the joint PDF f(x,y) = 8xy,0<x<y<1; 0 otherwise. Find

(i) marginal and conditional distributions

(ii) Check whether X and Y are independent . (8)

9. Given fxy (x, y) xy = cx(x-y), 0<x<2, -x<y<x

= 0, otherwise

(1) Evaluate C


(2) Find fx(x)

(3) fy/x(y/x) and

(4) fy(y) . (8)

10. Two random variables X and Y have the joint density f(x,y) = 2-x-y ; 0<x<1, 0<y<1

= 0, otherwise.

Show that Cov(X,Y) = .144

1−

(8)

11. Calculate the correlation coefficient for the following heights (in inches) of fathers X

and their sons Y.

X: 65 66 67 67 68 69 70 72

Y: 67 68 65 68 72 72 69 71 (8)

12.Two independent random variables X and Y are defined by

≤≤

=otherwise

xaxxf

,010,4

)( ≤≤

=otherwise

ybyyf

,010,4

)( .

Show that U =X + Y and V =X –Y are uncorrelated. (8)

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 th

169 of the variance of Y. Find the mean value of X and the

correlation co-efficient. (8)

14. If the equations of the two lines of regression of y on x and x on y are respectively

7x-16y+9=0; 5y-4x-3=0, calculate the coefficient of correlation. (8)

15. 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.

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)

16. Let (X,Y) be a two-dimensional non-negative continuous random variable having


the joint density f(x,y) =

≥≥+−

elsewhereyxxye yx

,00,0,4 )( 22

.Find the density function of 22 yxU += .(8)

17.If X and Y each follow an exponential distribution with parameter 1 and are independent , find the pdf of

U =X –Y. (8)

18.The lifetime of a certain brand of an electric bulb may be considered as a RV with mean 1200h and

standard deviation 250h. Find the probability , using central limit theorem, that the average lifetime of 60 bulbs

exceeds 1250h. (8)

19. A random sample of size 100 is taken from a population whose mean is 60and variance is 400. Using

Central limit theorem , with what probability can we assert that the mean of the sample will not differ from

μ = 60 by more than 4? (8)

20. A distribution with unknown mean μ has variance equal to 1.5. Use central limit theorem to find how large

a sample should be taken from the distribution in order that the probability will be atleast 0.95 that the

sample mean will be within 0.5 of the population mean. (8)

UNIT –III (CLASSIFICATION OF RANDOM PROCESSES)Part – A1.State the four types of a stochastic processes.2.Define a stationary process (or)strictly stationary process (or)strict sense stationary process. 3.Prove that a first order stationary has a constant mean.4.Consider the random process ( )θω += ttX 0cos)( , where θ is uniformly distributed in the interval π− to π . Check whether X(t) is stationary or not?5.When is a random process said to be ergodic.

6.Consider the Markov chain with tpm:

10003.01.04.02.0

007.03.0006.04.0

is it irreducible? If not find the class. Find the

nature of the states.7.Define Markov chain and one-step transition probability.8. State Chapman- Kolmogorow theorem.9.What is a Markov process?

10.If the transition probability matrix of a Markov chain is

21

21

10 , find the limiting distribution of the

chain. 11.State any two properties of a Poisson process.12. Prove that the difference of two independent poisson processes is not a poisson process.


13. If patients arrive at a clinic according to poisson process with mean rate of 2 per minute. Find the probability that during a 1-minute interval , no patients arrives.14.The probability that a person is suffering from cancer is 0.001. Find the probability that out of 4000 persons (a) Exactly 4 suffer because of cancer, (b) more than 3 persons will suffer from the disease.15.Define Sine-Wave Process.

Part- B

1. Give a random variable y with characteristic function φ (w) and a random process x(t) =cos( λ t +y). show

that {x( t)} is stationary in the wide sense ifφ (1) =0 and φ (2) =0. (8)

2.Show that the random process x(t )= Acos(wt +θ )is a wide sense stationary process if A and w are constants

and θ is a uniformly distributed random variable in (0,2π ). (6)

3.A random process X(t) defined by X(t) = Acost +Bsint , ∞<<∞− t , where A and B are independent

random variables each of which takes the value -2 with probability 1/3 and a value 1 with probability 2/3.

Show that X(t) is a WSS. (8)

4.The probability distribution of the process {x(t)}is given by

=+

=+==

+

−

0,1

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

)(

))((1

1

nat

at

nat

at

ntXPn

n

. Show

that it is not stationary. (10)

5. If x(t) = y coswt + z sinwt, where y and z are two independent normal RVS with

E(y) = E(z) = 0, E(y2) = E(z2) = 2σ and w is a constant, prove that {x( t)}is a SSS process of order 2. (8)

6.Show that the random process X(t)=Asin(wt+θ ) is WSS, A and w are constant and θ is uniformly

distributed in )2,0( π . (8)

7.Show that the random process x(t)= cos(t+φ ) where φ is uniformly distributed in (0,2π ) with probability

density function ππφ 20,

21)( <<= xxf is

(i). First order stationary.

(ii). Stationary is wide sense. (8)

(iii). Ergodic.

8. Distinguish between ‘stationary’ and weakly stationary stochastic processes. Given an example to each

type. Show that poisson process is an evolutionary process. (8)


9.The random binary transmission process {X(t)} is a WSS process with zero mean and autocorrelation

function T

Rτ

τ −= 1)( ,where T is a constant. Find the mean and variance of the time average of {X(t)} over

(0, T). Is {X(t)} mean Ergodic?. (8)

10.Prove that the random processes x(t) and Y(t) defined by X(t) = A cos 0ω t + B sin 0ω t ,

Y(t) = B cos 0ω t – A sin 0ω t are jointly wide – sense stationary, if A and B are uncorrelated zero mean

random variables with the same variance. (8)

11.The transition probability matrix of a Markov chain { }nX three states 1, 2 and 3 is

=

3.04.03.02.02.06.04.05.01.0

P and

the initial distribution isP(0)=(0.7,0.2,0.1). Find

(i). P{X2=3} and

(ii). P{X3=2, X2=3, X1=3, X0=2} . (8)

12.Three boys A,B, and C are throwing a ball to each other. A always throws ball to B and B always throws

the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process is Markovian to B

as to A. Find the transition matrix and classify the states. (8)

13.The man either drives a car or catches a train to go to the office each day. He never goes two 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 iff a

6 appeared . Find

(i)The probability that he takes a train on the third day.

(ii)The probability that he drives to work in the long run. (8)

14.Suppose that customers arrive at a bank according to a poisson process with a mean rate of 3 per minute.

Find the probability that during a time interval of 2 minutes

(i)Exactly 4customers arrive and (ii) more than 4 customers arrive. (8)

15. Derive the distributions of poisson process and find its mean and variance. (8)

16.The inter arrival time of a poisson process with parameter λ has an exponential distribution with ean λ1 (8)

17. If {X(t)} is a Gaussian process with μ(t) = 10 and C(t1,t2) = 16 21 tte −− find the probability that

(i) X(10) ≤ 8 and (ii) |X(10) – X(6)| ≤ 4. (8)

18. Write a critical note on ‘sine wave’ process and its applications. (8)


UNIT –IV (CORRELATION AND SPECTRAL DENSITIES)Part – A

1.Define autocorrelation function and prove that for a WSS process {X(t)}, )()( ττ XXXX RR =− .2.Stat any two properties of an auto correlation function.3. )(τXXR is an even function of τ .

Ie., )()( ττ XXXX RR =− .

4. The power spectral density of a random process {X(t)} is given by <

=elsewhere

S XX ,01,

)(ωπ

ω .Find its

autocorrelation function.

5. Find the variance of the stationary process {X(t)} whose ACF is given by 261916)(

ττ

++=R .

6. If the autocorrelation function of a stationary process is 231436)(

ττ

++=XXR , find the mean and variance

of the process. 7. Define Cross-correlation function and state any two of the properties.8. What is meant by spectral analysis?

9. The power spectral density of a WSS process is given by

>

≤−=aw

awwaab

wS;0

);()( . Find the

autocorrelation function of the process . 10. State any two uses of spectral density.

11. Given the power spectral density 241)(

ωω

+=XXS , find the average power of the process.

12. Give an example of Cross spectral density.13. Explain cross power spectrum.14. Determine the cross-correlation function corresponding to the cross-power density spectrum

2)(8)(jw

S XY +=

αω .

15.Check whether the following are valid autocorrelation function. (a) 2411)(sin2

τπ τ

+b

16.Write Wiener – Khintchine theorem.17. If τλτ 1)( −= eR is the autocorrelation function of a random process X(t), obtain the spectral density of X(t).

Part- B

1. Given that the autocorrelation function for a stationary ergodic process with no periodic components is

261425)(

ττ

++=R . Find the mean and varianc of the process {X(t)} . (8)


2. Derive the mean, autocorrelation and autocovariance of poisson process. (8)

3.The auto correlation function for a stationary process {X(t)} is given by ττ −+= eRXX 29)( .Find the mean of

the random variable ∫=2

0

)( dttXY and variance Of X(t). (8)

4. Consider two random processes X(t) = )cos(3 θω +t and Y(t) = )2cos(2 πθω −+t , where θ is a random

variable uniformly distributed in (0,2π ).Prove that )()0()0( τXYYYXX RRR ≥ . (8)

5. If X(t) is a WSS process and if Y(t) = X(t+a) - X(t-a), Prove that

RYY(τ ) =2Rxx (τ ) – Rxx (τ +2a) – Rxx (τ -2a). (8)

6.Find the auto correlation function of the periodic time function {X(t)} = tA ωsin . (8)

7.Given a stationary random process )100cos(10)( θ+= ttX ,Where ),( ππθ −∈ followed uniform distribution

Find the auto correlation function of the process. (8)

8. {X(t)} and {Y(t)} are zero mean and stochastically independent random process having auto correlation

functions ττ −= eRXX )( and π ττ 2cos)( =YYR respectively. Find (i)The auto correlation function of

W(t)= X(t)+ Y(t) and Z(t) =X(t)-Y(t).(ii)The cross correlation function of W(t) and Z(t). (8)

9.The power spectrum of a WSS process X = {x(t)} is given by 22 )1(1)(w

wS+

= .Find

the auto – correlation function and average power of the process. (8)

10. The Power spectral density of a zero mean wide stationary process X(t) is given by

<

=otherwise

wwkwS

,0,

)( 0

,where k is a constant. Show that X(t) and X(t + 100

π) are uncorrelated . (8)

11.Calculate the power spectral density of a stationary random process whose auto

correlation is ττ aXX eR −=)( . (8)

12.Given the power spectral density of a continuous process as 45

9)( 24

2

+++=ww

wwS XX .

Find the mean square value of the process. (8)

13. If the cross – correlation of two processes {X(t)} and {Y(t)} is

2),( ABttRXY =+ τ [ sin (woτ ) + cos (wo(2t +τ ) ] ,where A ,B are wo are constants. Find

the cross power spectrum. (8)


14.The cross power specrum of real random process X(t) and Y(t) is given by <+

=otherwise

wjbwawS XY ,0

1,)( .

Find the cross correlation function. (8)

15.The auto correlation function of the random binary transmission X(t) is given by

>

<−=T

TTR

τ

ττ

τ,0

,1)( .

Find the power spectrum of the process X(t). (8)

16.The auto correlation function of the poisson increment process is given by

≤

>

−+

=ετλ

ετετ

ελλ

τ,

,1)(

2

2

R

.Prove that its spectral density is given by 22

2

2 2sin4

)(2)(w

w

wwSε

ελδπ λ

+= .(8)

17.The power spectral density of a WSS process is given by

>

≤−=aw

awwaab

wS,0

),()( .

Find auto-correlation function of the process . (8)

18.State and prove weiner-khintchine theorem. (8)

19. The Autocorrelation of a random telegraph signal process is given by

τατ 22)( −= eAR Determine the power spectral density random telegraph signal. (8)

UNIT –V

(LINEAR SYSTEMS WITH RANDOM INPUTS)Part – A1.describe a linear system.2.Define a system. When is it called linear system?3.State the properties of a linear filter.4.Describe a linear system with an random input.5. Give an example for a linear system.6. Define Time invariance.7. State Causality.8. State stable.9. Define White Noise (or) Gaussian Noise.


10.Define Thermal Noise.11.Define Band-Limited White Noise.12.Define Filters.13.Find the auto correlation function of Gaussian white Noise. 14. State auto correlation function of Gaussian white noise.15.Find the system transfer function , if a linear time invariant system has an impulse function

≥

≤=ct

ctcth,0

,21

)( .

16.A white signal with PSD 2

η is applied to an RC LPF. Find the auto correlation of the O/P signal of the

filter.

Part- B

1.Consider the white Gaussian Noise of zero mean and power spectral density 2

0N applied to a low pass cR

filter where transfer function is ifRcfH

π211)(

+= . Find the output spectral density and auto

correlationfunction of the output process. (8)

2.Show that Syy(w) = )()( 2 WSWH xx , where Sxx(w) and Syy(w) are the power spectral density functions of

the input X(t) and the output Y(t) and H(w) is the system transfer function. (8)

3.If the input X(t) and its output Y(t) are related by Y (t) = ∫∞

∞−

− duutxuh )()( ,then prove

that the system is a linear time – invariant system. (8)

4.If the input to a time-invariant, stable linear system is a WSS process, then prove that

the output will also be WSS process . (8)

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 0=xµ and Rxx (τ ) = τ2−e .Find yµ , Sxx (w) and Syy(w) if the system function is given by

H (w) = iW 2

1+

.(8)

6.If the input X(t) and the output Y(t) are connected by the differential equation )()()( txtydt

tdyT =+ .Prove

that they can be related by means of a convolution type integral. Assume that x(t) and y(t) are zero for t ≤ 0(8)


7.A Circuit has an impulse response given by h(t) =

≤≤ TtT

elsewhere

0,1,0

.Evaluate Syy (w) in

terms of Sxx (w). (8)

8.X(t) is the input voltage and Y(t) is the output voltage also {X(t)} is a stationary process with x = 0 and

Rxx(τ ) = τα−e . Find yµ , Syy(w), Ryy(τ ), if the power transfer function is iLWRRWH

+=)( .(8)

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

H(W)=1 for 00 ωωω <<− .If the autocorrelation function of the input process is )(2

0 tN

δ , find the

autocorrelation function of the output process. (8)

10.Consider a system with transfer functioniw+1

1 .An input signal with autocorrelation function 2)( mtm +δ

is fed as input to the system. Find the mean and mean square value of the output. (8)

11.A linear system is described by the impulse response ( )Rct

eRc

th −= 1)( .Assume an input signal whose

autocorrelation function is )(τδB .Find the autocorrelation, mean and power of the output. (8)

12.If {N(t)} is a band limited white noise centered at a carrier frequency 0ω such that

<−=

elsewhere

NS B

NN,0

,2)( 0

0 ωωωω . Find the autocorrelation of {N(t)}. (8)

13.For a linear system with random input x(t), the impulse response h(t) and output y(t), obtain the cross correlation function )(τyxR and the output autocorrelation function )(τyyR .(8) 14.A random process X(t) having the autocorrelation function τατ −= peRxx )( , where p and α are real

positive constants, is applied to the input of the system with impulse response

<>

=−

0,00,

)(t

teth

tλλ, where λ is

a real positive constant. Find the autocorrelation function of the network’s response y(t). (8) 15. If )()cos()( 0 tNtAtY ++= θω ,where A is a constant, θ is a random variable with a uniform distribution

in ),( ππ− and {N(t)} is a band limited Gaussian white noise with a power spectral density


<−=

elsewhere

NS B

NN,0

,2)( 0

0 ωωω . Find the power spectrum density of {Y(t)} . Assume that N(t) and θ are

independent . (8)

16. If {X(t)} is a band limited process such that σωω >= ,0)(xxS prove that

[ ] )0()()0(2 22XXXXXX RRR τστ ≤− .(8)

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

h(t) = 0, ≥− te t for an input with power density spectrum ∞<<− ∞ f,2

0η.(8)

18. A WSS process {X(t)} with τατ −= AeR )( , where A and α are real positive constants is applied to the

input of a linear time invariant system with h(t) = bte − U(t), where b is a real positive constant. Find the power

spectral density of the output of the system. (8)

19.Define white noise. Find the Autocorrelation function of the white noise . (8)

.

probability random process qb 2

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