I (Random Variables - Hariganesh.com

Engineering Mathematics

2016

SUBJECT NAME

: Probability & Random Processes es

SUBJECT CODE

: MA6451

MATERIAL NAME

: University Questions

MATERIAL CODE

: JM08AM1004

REGULATION

: R2013

UPDATED ON

: February 2016

(Upto N/D 2015 201 Q.P)

(Scan the above Q.R code for the direct download of this material)

Name of the Student:

Branch:

Unit – I (Random Random Variables) Variables • Problems on Discrete & Continuous R.Vs 1.

A random variable X has the following probability distribution. X 0 P(x) 0

1

2

3

4

5

6

7

k

2k

2k

3k

k2

2k 2

7k 2 + k

Find: (1) The value of k (2) P (1.5 < X < 4.5 / X > 2) and (3) The smallest value of n for which P ( X ≤ n ) >

1 . 2

(N/D 2010),(M/J (M/J 2012) 2012),(M/J 2014) 2.

Show that for the probability function

1  , x = 1, 2, 3...  p( x ) = P ( X = x ) =  x ( x + 1 ) E ( X ) does not exist. (N/D 2012)  0, otherwise  3.

The probability function of an infinite discrete distribution is given by

P( X = j) =

1 ( j = 1, 2, 3, ...) Find 2j

(1) Mean of X

(2) P ( X is even) and (3) P ( X is divisible by 3) (N/D 2011)

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

Page 1

Engineering Mathematics 4.

2016

The probability mass function of random variable X is defined as P ( X = 0) = 3C 2 ,

P ( X = 1) = 4C − 10C 2 , P ( X = 2) = 5C − 1 , where C > 0 and P ( X = r ) = 0 if r ≠ 0,1, 2 . Find (1) The value of C (2) P (0 < X < 2 / x > 0) (3) The distribution function of X (4) The largest value of X for which F ( x )
0 A random variable X has pdf f ( x ) =  . Find the rth moment of otherwise  0, (M/J 2013) X about origin. Hence find the mean and variance. The probability density function of a random variable X is given by

0< x0 12. Find MGF corresponding to the distribution f (θ ) =  2 and hence find  0, otherwise  its mean and variance.

(N/D 2012)

• Problems on Distributions 1.

Derive Poisson distribution from binomial distribution.

(N/D 2013),(N/D 2014)

2.

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?

3.

State and prove memory less property of Geometric distribution.

4.

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

P ( X < 3 ) , P ( X > 7 ) , P ( 2 < X < 5 ) and P ( X = 7 ) .

5.

(A/M 2010) (N/D 2015)

(M/J 2013)

The time in hours required to repair a machine is exponentially distributed with parameter λ = 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?

(M/J 2012)

6.

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? (A/M 2010),(A/M 2011)

7.

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


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2016

(1) at least 44.5 cc/min (2) at most 35.0 cc/min (3) any where from 30.0 to 40.0 cc/mm.

(N/D 2012)

8.

In a normal distribution, 31% of items are under 45 and 8% of items are over 64. Find the mean and the standard deviation of the distribution. (N/D 2015)

9.

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? (N/D 2011)

10. Given that X is distributed normally, if P ( X < 45) = 0.31 and P ( X > 64) = 0.08 , find the mean and standard deviation of the distribution.

(M/J 2012)

(

11. If X and Y are independent random variables following N (8, 2) and N 12, 4 3 respectively, find the value of λ such that P [ 2 X − Y ≤ 2λ ] = P [ X + 2Y ≥ λ ] .

)

(N/D 2010)

Unit – II (Two Dimensional Random Variables) • Joint distributions – Marginal & Conditional 1.

The joint probability mass function of ( X , Y ) is given by p ( x , y ) = k ( 2 x + 3 y ) ,

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 ) . (N/D 2013),(N/D 2014) 2.

The bivariate probability distribution of ( X , Y ) given below: Y 1 2 3 4 5 6 X 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. (A/M 2010)


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The joint probability mass function of

2016

( X , Y ) is given by p ( x , y ) =

1 (2x + 3 y); 72

x = 0,1, 2 and x = 1, 2, 3 . Find all the marginal and conditional probability function of X and Y . (N/D 2015) 4.

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

8 xy , 9

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 ) . 5.

(A/M 2010)

The joint probability density function of random variable X and Y is given by

 8 xy , 1≤ x ≤ y ≤ 2  f ( x, y) =  9 . Find the conditional density functions of X and Y .  0, otherwise (N/D 2011) 6.

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

x2 f ( x , y ) = xy + , 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 . Compute P (Y < 1 / 2) , 8 P ( X > 1 / Y < 1 / 2) and P ( X + Y ≤ 1) . 2

7.

(N/D 2012)

 Cx ( x − y ), 0 < x < 2, − x < y < x . otherwise  0,

Given the joint pdf of X and Y f ( x , y ) = 

(1) Evaluate C (2) Find the marginal pdf of X (3) Find the conditional density of Y / X . (M/J 2013) 8.

The joint pdf of ( X , Y ) is f ( x , y ) = e −

−( x+ y )

; x , y ≥ 0 . Are X and Y independent? (N/D 2015)

9.

If the joint pdf of two dimensional random variable ( X , Y ) is given by

 2 xy , 0 < x < 1; 0 < y < 2 x + f ( x, y ) =  . Find 3  0, otherwise

 

(i) P  X >

(M/J 2014)

1 2 

(ii) P ( Y < X )


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2016

(iii) P [ X + Y ≥ 1] (iv) Find the conditional density functions.

• Covariance, Correlation and Regression 1.

The joint pdf of the random variable ( X , Y ) is f ( x , y ) = 3 ( x + y ) ,

0 ≤ x ≤ 1, 0 ≤ y ≤ 1 , x + y ≤ 1 find Cov ( X , Y ) . 2.

(M/J 2014)

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 ≤ 1 and f ( x , y ) = 0 , otherwise. (A/M 2010) 3.

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

f ( x , y ) = Kxye

(

− x2 + y2

) , x > 0, y > 0 . Find the value of K and Cov X , Y . Are X ( )

and Y independent? 4.

The joint pdf of a random variable ( X , Y ) is f ( x ) = 25e −5 y ; 0 < x < 0.2, y > 0 . Find the covariance of X and Y .

5.

(M/J 2012)

(N/D 2015)

The joint probability density function of the two dimensional random variable ( X , Y )

 2 − x − y , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 . Find the correlation coefficient otherwise  0,

is f ( x , y ) = 

between X and Y . 6.

(N/D 2011)

Two random variables X and Y have the joint probability density function given by

 k (1 − x 2 y ), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 f XY ( x , y ) =  . otherwise  0,

(N/D 2010)

(1) Find the value of ‘ k ’ (2) Obtain the marginal probability density functions of X and Y . (3) Also find the correlation coefficient between X and Y . 7.

Two independent random variables X and Y are defined by

 4ax , 0 < x < 1  4by , 0 < y < 1 f X ( x) =  and fY (y) =  . Show that otherwise otherwise  0,  0, (M/J 2013) U = X + Y and V = X − Y are uncorrelated.


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8.

If X and Y are uncorrelated random variables with variances 16 and 9. Find the correlation co-efficient between X + Y and X − Y . (M/J 2012)

9.

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 ) . (N/D 2012)

10. The equations of two regression lines are 3 x + 12 y = 19 and 3 y + 9 x = 46 . Find x , y and Correlation Coefficient between X and Y .

(M/J 2013)

11. The regression equation of X on Y is 3Y − 5 X + 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. (A/M 2011) 12. 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 x = 55 .

(M/J 2014)

• Transformation of the random variables 1. If X and Y are independent random variables with density function

y 1, 1 ≤ x ≤ 2  , 2≤ y≤4 and fY ( y ) =  6 , find the density function of f X ( x) =   0, otherwise  0, otherwise Z = XY . 2.

(A/M 2011)

e − x , x ≥ 0 and X and Y are independent with a common PDF (exponential): f ( x ) =   0, x < 0

e − y , y ≥ 0 f ( y) =  . Find the PDF for X − Y . (N/D 2011)  0, y < 0 3. The random variables X and Y each follow exponential distribution with parameter 1 and are independent. Find the pdf of U = X − Y . (N/D 2015) −x

−y

4. If X and Y are independent RVs with pdf’s e − , x ≥ 0 and e − , y ≥ 0 respectively, find the pdfs of U =

X and V = X + Y . Are U and V independent? X +Y

(N/D 2013)

5. If X and Y are independent random variables with probability density functions

f X ( x ) = 4e −4 x , x ≥ 0; fY ( y ) = 2e −2 y , y ≥ 0 respectively.


(N/D 2012)

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(i) Find the density function of U =

X , V = X +Y X +Y

(ii) Are U and V independent? (iii) What is P ( U > 0.5 ) ? 6. 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 ≤ 1 . Find the p.d.f of U = XY . (M/J 2012) 7. If X and Y are independent RVs each normally distributed with mean zero and variance σ 2 , find the pdf of R =

Y X 2 + Y 2 and φ = tan −1  X

 . 

(N/D 2013)

8. If X and Y are independent continuous random variables, show that the pdf of

U = X + Y is given by h( u) =

∞

∫

f x (v ) f y ( u − v )dv .

(N/D 2010)

−∞

Unit – III (Random Processes) • Verification of SSS and WSS process 1.

Examine whether the random process { X ( t )} = A cos(ω t + θ ) is a wide sense stationary if A and ω are constants and θ is uniformly distributed random variable in (0,2π). (A/M 2010),(N/D 2011)

2.

A random process X ( t ) defined by X ( t ) = A cos t + B sin t , − ∞ < 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. 3.

(A/M 2011),(M/J 2013),(N/D 2015)

Prove that the random processes X ( t ) and Y (t ) defined by

X ( t ) = A cos ω t + B sin ω t and Y ( t ) = B cos ω t − A sin ω t are jointly wide sense stationary. 4.

(M/J 2014)

If the two RVs Ar and Br are uncorrelated with zero mean and

E ( Ar2 ) = E ( Br2 ) = σ r2 , show that the process x ( t ) = ∑ ( Ar cos ω r t + Br sin ω r t ) is n

r =1

wide-sense stationary.


(N/D 2013),(N/D 2014)

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The process { X ( t )} whose probability distribution under certain condition is given by

 (at )n−1 , n = 1, 2...  (1 + at )n+1 . Find the mean and variance of the process. P { X ( t ) = n} =   at , n=0  1 + at Is the process first-order stationary? (N/D 2010),(N/D 2011),(N/D 2012),(M/J 2014),(N/D 2014) 6.

If { X ( t )} is a WSS process with autocorrelation R(τ ) = Ae order moment of the RV { X (8) − X (5)} .

−α τ

, determine the second (M/J 2012)

• Problems on Markov Chain 1. The transition probability matrix of a Markov chain { X ( t )} , n = 1, 2, 3, ... having three

 0.1 0.5 0.4    states 1, 2, 3 is P =  0.6 0.2 0.2  , and the initial distribution is  0.3 0.4 0.3    P (0) = [ 0.7 0.2 0.1] , Find P ( X 2 = 3 ) and P ( X 3 = 2, X 2 = 3, X 1 = 3, X 0 = 2 ) . (A/M 2010) 2. A man either drives a car or catches a train to go to office each day. He never goes two days in a row by train. But he drives one day, then the next day 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 dice and drove to work if and only if a 6 appeared. Find the probability that he takes a train on the fourth day and the probability that he drives to work on the fifth day. (N/D 2015)

• Poisson Process 1.

If the process { X ( t ); t ≥ 0} is a Poisson process with parameter λ , obtain

P [ X (t ) = n] . Is the process first order stationary? (N/D 2010),(N/D 2012),(M/J 2014)

2.

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. (N/D 2011) Define a Poisson process. Show that the sum of two Poisson processes is a Poisson process. (M/J 2013),(N/D 2013)


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4.

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 that 1 minute (2) between 1 minute and 2 minute and (3) 4 min. or less. (M/J 2012)

5.

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 second interval (2) The probability that the number of message arrivals in an interval of duration 5 seconds is between 3 and 7. (A/M 2010)

6.

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 min. (1) exactly 4 customers arrive and

7.

(2) more than 4 customers arrive

(N/D 2013)

(3) Fewer than 4 customers arrive

(N/D 2015)

Prove that the interval between two successive occurrences of a Poisson process with parameter λ has an exponential distribution with mean

1

λ

.

(A/M 2011)

• Normal (Gaussian) & Random telegraph Process 1.

If { X ( t )} is a Gaussian process with µ ( t ) = 10 and C ( t1 , t 2 ) = 16e

− t1 − t 2

, find the

probability that (1) X (10) ≤ 8 (2) X (10) − X (6) ≤ 4 2.

(A/M 2011),(N/D 2013),(N/D 2014)

Suppose that X ( t ) is a Gaussian process with µ x = 2, Rxx (τ ) = 5e probability that X (4) ≤ 1 .

−0.2 τ

. Find the (M/J 2012)

3.

Define a semi random telegraph signal process and prove that it is evolutionary. (M/J 2013),(N/D 2015)

4.

Define random telegraph signal process and prove that it is wide-sense stationary. (N/D 2013)


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5.

Define a semi random telegraph signal process and random telegraph signal process and prove also that the former is evolutionary and the latter is wide-sense stationary. (N/D 2014)

6.

Prove that a random telegraph signal process Y ( t ) = α X ( t ) is a Wide Sense Stationary Process when α is a random variable which is independent of X ( t ) , assume value −2 λ t1 − t2

−1 and +1 with equal probability and RXX ( t1 , t 2 ) = e −

.

(N/D 2010),(N/D 2012),(M/J 2014)

Unit – IV (Correlation and Spectral densities) • Auto Correlation from the given process 1.

2.

Find the autocorrelation function of the periodic time function of the period time function { X ( t )} = A sin ω t .

(A/M 2010)

Find the mean and auto correlation of the Poisson process.

(M/J 2014)

• Relationship between 1.

RXX (τ )

and

S XX ( ω )

Define spectral density of a stationary random process X ( t ) . Prove that for a real random process X ( t ) , the power spectral density is an even function. (M/J 2013)

2.

The autocorrelation function of the random binary transmission { X ( t )} is given by

R(τ ) = 1 −

τ T

for τ < T and R(τ ) = 0 for τ < T . Find the power spectrum of the

process { X ( t )} . 3.

(A/M 2010)

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

1 − τ , for τ ≤ 1

is R(τ ) = 

 0,

4.

elsewhere

.

(N/D 2010),(N/D 2012)

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

R XX (τ ) =

A2 cos ( ω 0τ ) . 2


(M/J 2012)

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A random process { X ( t )} is given by X ( t ) = A cos pt + B sin pt , where A and B are independent random variables such that E ( A) = E ( B ) = 0 and

E ( A2 ) = E ( B 2 ) = σ 2 . Find the power spectral density of the process. (N/D 2014) 6.

The autocorrelation function of a random process is given by

λ 2 ;τ > ε  . Find the power spectral density of the process. R(τ ) =  2 λ  τ  λ +  1 −  ; τ ≤ ε ε ε   (N/D 2011) 7.

The autocorrelation function of a random process is given by

λ 2 ;τ > ε  . Prove that its spectral density is R(τ ) =  2 λ  τ  λ +  1 −  ; τ ≤ ε ε ε   S ( ω ) = 2πλ δ (δ ) + 2

8.

(

4λ sin 2 ω ∈ ∈ ω 2

2

)

2 .

The Auto correlation function of a WSS process (random telegraph single process) is given by R(τ ) = α 2 e

−2 λ τ

, determine the power spectral density of the process

(random telegraph single process). 9.

(N/D 2013)

(A/M 2011),(N/D 2015)

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

R(τ ) = e−ατ . 2

(N/D 2014)

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

R(τ ) = α 2 e

−2 τ

. Determine the power density spectrum of the random telegraph

signal.

(N/D 2013)

11. Find the power spectral density of a WSS process X ( t ) which has an autocorrelation

Rxx (τ ) = A0 1 − τ / T  , − T ≤ t ≤ T .

(N/D 2012)

12. Find the autocorrelation function of the process { X ( t )} for which the power spectral density is given by S XX (ω ) = 1 + ω 2 for ω < 1 and S XX (ω ) = 0 for ω > 1 .(A/M 2010)


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

 1, ω < ω0 π  S (ω ) =  . Find R (τ ) and show that X ( t ) and X  t +  are  ω0  0, otherwise uncorrelated.

(A/M 2011)

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

b  (a − ω ) , ω ≤ a S (ω ) =  a . Find the autocorrelation function of the process. 0, ω >a  (N/D 2013),(N/D 2014) 15. The power spectrum of a WSS process { X ( t )} is given by S (ω ) =

(1 + ω 2 )

auto correlation function R(τ ) .

• Relationship between

1

2

. Find its

(N/D 2015)

RXY (τ )

and

S XY ( ω )

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

{

}

AB sin(ω 0τ ) + cos ω 0 ( 2t + τ )  where A, B and ω0 are constants. 2 (M/J 2012) Find the cross-power spectrum S XY (ω ) . RXY ( t , t + τ ) =

2. The cross – power spectrum of real random processes { X ( t )} and {Y ( t )} is given by

a + bjω , for ω < 1 S xy (ω ) =  . Find the cross correlation function. elsewhere 0, (N/D 2010),(A/M 2011),(N/D 2011) 3. If the cross power spectral density of X (t ) and Y (t ) is

ibω  , −α < ω < α, α > 0 a + where a and b are constants. Find the S XY (ω ) =  α  0, otherwise cross correlation function.

(M/J 2013)

4. Two random processes X ( t ) and Y( t ) are defined as follows: X ( t ) = Acos(ω t + θ ) and Y( t ) = B sin(ω t + θ ) where A, B and ω are constants; θ is a uniform random


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variable over ( 0, 2π ) . Find the cross correlation function of X ( t ) and Y( t ) . (M/J 2013),(N/D 2015)

• Properties, Theorem and Special problems 1. State and prove Weiner – Khintchine Theorem. (N/D 2010),(A/M 2011),(N/D 2011),(N/D2012),(M/J 2013),(M/J 2014) 2. If { X ( t )} and {Y ( t )} are two random processes with auto correlation function

RXX (τ ) and RYY (τ ) respectively then prove that RXY (τ ) ≤ RXX (0) RYY (0) . Establish any two properties of auto correlation function RXX (τ ) .(N/D 2010),(N/D2012) 3. Given the power spectral density of a continuous process as S XX ( ω ) =

ω2 + 9 . ω 4 + 5ω 2 + 4

Find the mean square value of the process. (N/D 2011) 4. A stationary random process X ( t ) with mean 2 has the auto correlation function −τ

1

RXX (τ ) = 4 + e 10 . Find the mean and variance of Y = ∫ X ( t ) dt .

(M/J 2012)

0

5. The random binary transmission process { X ( t )} is a WSS process with zero mean and autocorrelation function R(τ ) = 1 −

τ

, where T is a constant. Find the mean and T variance of the time average of { X ( t )} over ( 0,T ) . Is { X ( t )} mean ergodic? (N/D 2014) 6.

{ X ( t )} and {Y ( t )} are zero mean and stochastically independent random processes having autocorrelation functions RXX (τ ) = e

−τ

and RYY (τ ) = cos 2πτ 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 ) .

(A/M 2010)

7. Let X ( t ) and Y ( t ) be both zero-mean and WSS random processes Consider the random process Z ( t ) defined by Z ( t ) = X ( t ) + Y ( t ) . Find (1) The Auto correlation function and the power spectrum of Z ( t ) if X ( t ) and

Y ( t ) are jointly WSS.


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(2) The power spectrum of Z ( t ) if X ( t ) and Y ( t ) are orthogonal. (M/J 2012) 8. If Y ( t ) = X ( t + a ) − X ( t − a ) , prove that RYY (τ ) = 2 RXX (τ + 2a ) − RXX (τ − 2a ) . Hence prove that SYY (ω ) = 4 sin 2 aω S XX (ω ) .

(N/D 2015)

9. If the process { X ( t )} is defined as X ( t ) = Y ( t ) Z ( t ) where {Y ( t )} and { Z( t )} are independent WSS processes, prove that (1) Rxx (τ ) = Ryy (τ ) Rzz (τ ) and

1 (2) S xx ( ω ) = 2π

∞

∫ S ( α ) S ( ω − α ) dα yy

(N/D 2013)

zz

−∞

Unit – V (Linear Systems with Random inputs) • Input and Output Process 1.

2.

If the input to a time invariant stable, linear system is a WSS process, prove that the output will also be a WSS process. (N/D 2011),(M/J 2013) Show that if the input { X ( t )} is a WSS process for a linear system then output {Y ( t )} is a WSS process. Also find RXY (τ ) .

3.

(N/D 2010),(N/D 2012),(M/J 2014)

For a input – output linear system ( X ( t ), h( t ), Y ( t ) ) , derive the cross correlation function RXY (τ ) and the output autocorrelation function RYY (τ ) .

4.

(N/D 2011)

Check whether the following systems are linear (1) y ( t ) = t x ( t ) (2) y( t ) = x 2 ( t ) . (N/D 2014)

5.

Prove that the spectral density of any WSS process is non-negative.

6.

Consider a system with transfer function

(N/D 2013)

1 . An input signal with autocorrelation 1 + jω

function mδ (τ ) + m 2 is fed as input to the system. Find the mean and mean-square value of the output.


(A/M 2011),(M/J 2012)

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If { X ( t )} is a WSS process and if Y ( t ) =

7.

∞

∫ h(ξ ) X (t − ξ )dξ

then prove that

−∞

(1) RXY (τ ) = RXX (τ )* h(τ ) where * stands for convolution. (2) S XY (ω ) = S XX (ω ) H * (ω ) . If { X ( t )} is a WSS process and if Y ( t ) =

8.

(M/J 2012) ∞

∫ h( u) X (t − u)du , prove that:

−∞

(i) RXY (τ ) = RXX (τ ) ∗ h( −τ ) (ii) RYY (τ ) = RXY (τ ) ∗ h(τ ) where ∗ denotes convolution (iii) S XY (ω ) = S XX (ω ) H ∗ (ω ) where H ∗ (ω ) is the complex conjugate of H (ω ) (iv) SYY (ω ) = S XX (ω ) H (ω )

2

(N/D 2015)

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

9.

H (ω ) = 1 for −ω0 < ω < ω0 . If the autocorrelation function of the input process is

N0 δ ( t ) , find the autocorrelation function of the output process. 2 10. If X ( t ) is the input voltage to a circuit and Y ( t ) is the output voltage. stationary random process with µ X = 0 and RXX (τ ) = e

−2 τ

(A/M 2010)

{ X (t )} is a

. Find the mean µY and

power spectrum SYY (ω ) of the output if the system transfer function is given by H (ω ) = 11.

1 . ω + 2i

(N/D 2010),(N/D 2012)

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 µ x = 0 and Rxx (τ ) = e

Ryy (τ ) , if the power transfer function is H ( ω ) =

−α τ

. Find µ y , S yy ( ω ) and

R . (N/D 2013),(M/J 2014) R + iLω

• Input and Output Process with Impulse Response 1.

A system has an impulse response h( t ) = e − β t U ( t ) , find the power spectral density of the output Y ( t ) corresponding to the input X ( t ) .

(N/D 2010),(N/D 2012),(M/J 2014)


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2016

A random process X ( t ) is the input to a linear system whose impulse function is −2 τ

h( t ) = 2e − t ; t ≥ 0 . The auto correlation function of the process is RXX (τ ) = e − τ . Find the power spectral density of the output process Y ( t ) . 3.

A random process

(M/J 2013)

X ( t ) is the input to a linear system whose impulse response is

h( t ) = 2e − t , t ≥ 0 . If the autocorrelation function of the process is RXX (τ ) = e −2 τ , determine the cross correlation function output process

RXY (τ ) between the input process X ( t ) and the

Y (t ) and the cross correlation function RYX (τ ) between the output process

Y ( t ) and the input process X ( t ) . 4.

(N/D 2015)

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

RXX (τ ) = Aδ (τ ) is applied to a linear system at time t = 0 where f (τ ) represent the − bt

impulse function. The linear system has the impulse response of h( t ) = e − u( t ) where

u( t ) represents the unit step function. Find RYY (τ ) . Also find the mean and variance of Y (t ) . 5.

(A/M 2011),(M/J 2012)

A wide sense stationary random process { X ( t )} with autocorrelation RXX (τ ) = e

−a τ

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 − bt u( t ) where b is a real positive constant. Find the autocorrelation of the output Y ( t ) of the system.(A/M 2010) 6.

−t 1 RC e u( t ) . Assume an RC input process whose Auto correlation function is Bδ (τ ) . Find the mean and Auto correlation function of the output process. (A/M 2011)

A linear system is described by the impulse response h( t ) =

 t 

7.

1 −  RC  A linear system is described by the impulse response h( t ) = . Assume an e RC input signal whose autocorrelation function is Bδ (τ ) . Find the autocorrelation mean and power of the output.

8.

(N/D 2014)

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 S yy (ω ) = H (ω ) S xx (ω ) where H (ω ) 2

is Fourier transform of h( t ) .


(N/D 2011),(M/J 2013)

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2016

• Band Limited White Noise 1.

A wide sense stationary noise process N (t ) has an auto correlation function

RNN (τ ) = Pe −3 τ where P is a constant. Find its power spectrum. 2.

(M/J 2013)

If Y ( t ) = A cos(ω0 t + θ ) + 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

 N0 , for ω − ω 0 < ω B  . Find the power with power spectral density S NN (ω ) =  2  0, elsewhere spectral density {Y ( t )} . Assume that { N ( t )} and θ are independent. (N/D 2010),(N/D 2012),(N/D 2013),(M/J 2014) 3.

If Y ( t ) = A cos(ω t + θ ) + 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

N0 for ω − ω0 < ω B and S NN (ω ) = 0 , elsewhere. 2 Find the power spectral density of Y ( t ) , assuming that N ( t ) and θ are independent.

a power spectral density S NN (ω ) =

(A/M 2010) 4.

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

 N0 , for ω − ω0 < ω B  . Find the autocorrelation of { N ( t )} . S NN (ω ) =  2  0, elsewhere (A/M 2011),(M/J 2012) 5.

If { X ( t )} is a band limited process such that S XX (ω ) = 0 when ω > σ , prove that

2 RXX (0) − RXX (τ ) ≤ σ 2τ 2 RXX (0) . 6.

A white Gaussian noise X ( t ) with zero mean and spectral density

(A/M 2010)

N0 is applied to a 2

low-pass RC filter shown in the figure.


Page 19


Determine the autocorrelation of the output Y ( t ) .

2016

(N/D 2011)

----All the Best----


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