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MA2261

PROBABILITY AND RANDOM PROCESSES

C.Ganesan, M.Sc., M.Phil.,

Assistant Professor of Mathematics

Dhanalakshmi College of Engineering

Mobile: 9841168917

Website: www.hariganesh.com

MA2261 - PROBABILITY AND RANDOM PROCESSES

UNIT.1 RANDOM VARIABLES

Discrete and continuous random variables – Moments - Moment generating functions and their properties. Binomial, Poisson ,Geometric, Uniform, Exponential, Gamma and normal distributions –Function of Random Variable.

UNIT.2 TWO DIMENSIONAL RANDOM VARIBLES

Joint distributions - Marginal and conditional distributions – Covariance - Correlation and Regression - Transformation of random variables - Central limit theorem (for iid random variables)

UNIT.3 CLASSIFICATION OF RANDOM PROCESSES

Definition and examples - first order, second order, strictly stationary, wide-sense stationary and ergodic processes - Markov process - Binomial, Poisson and Normal processes - Sine wave process – Random telegraph process.

UNIT.4 CORRELATION AND SPECTRAL DENSITIES

Auto correlation - Cross correlation - Properties – Power spectral density – Cross spectral density -Properties – Wiener-Khintchine relation – Relationship between cross power spectrum and cross correlation function.

UNIT.5 LINEAR SYSTEMS WITH RANDOM INPUTS

Linear time invariant system - System transfer function – Linear systems with random inputs – Auto correlation and cross correlation functions of input and output – white noise.

Text Book

1. Oliver C. Ibe, “Fundamentals of Applied probability and Random processes”, Elsevier, First Indian Reprint ( 2007) (For units 1 and 2)

2. Peebles Jr. P.Z., “Probability Random Variables and Random Signal Principles”, Tata McGraw-Hill Publishers, Fourth Edition, New Delhi, 2002. (For units 3, 4 and 5).

References

1. Miller,S.L and Childers, S.L, “Probability and Random Processes with applications to Signal Processing and Communications”, Elsevier Inc., First Indian Reprint 2007.

2. H. Stark and J.W. Woods, “Probability and Random Processes with Applications to Signal Processing”, Pearson Education (Asia), 3rd Edition, 2002.

3. Hwei Hsu, “Schaum’s Outline of Theory and Problems of Probability, Random Variables and Random Processes”, Tata McGraw-Hill edition, New Delhi, 2004.

4. Leon-Garcia,A, “Probability and Random Processes for Electrical Engineering”, Pearson Education Asia, Second Edition, 2007.

5. Yates and D.J. Goodman, “Probability and Stochastic Processes”, John Wiley and Sons, Second edition, 2005.

Engineering Mathematics 2013

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1

SUBJECT NAME : Probability & Random Process

SUBJECT CODE : MA 2261

MATERIAL NAME : University Questions

MATERIAL CODE : JM08AM1004

Name of the Student: Branch:

Unit – I (Random Variables)

Problems on Discrete & Continuous R.Vs

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 2k 2

2k 27k k

Find:

(1) The value of k

(2) (1.5 4.5 / 2)P X X and

(3) The smallest value of n for which 1

( )2

P X n .

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

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

2( 1) 4 10P X C C , ( 2) 5 1P X C , where 0C and

( ) 0P X r if

0,1, 2r . Find

(1) The value of C

(2) (0 2 / 0)P X x

(3) The distribution function of X

(4) The largest value of X for which1

( )2

F x . (A/M 2010)

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



, 0 1

( ) (2 ), 1 2

0, otherwise

X

x x

f x k x x

.

(1) Find the value of ‘ k ’.

(2) Find (0.2 1.2)P x

(3) What is 0.5 1.5 / 1P x x

(4) Find the distribution function of ( )f x . (A/M 2011)

4. A continuous R.V. X has the p.d.f. 2,

( ) 1

0, elsewhere

kx

f x x

. Find

(1) the value of k

(2) Distribution function of X

(3) ( 0)P X (N/D 2011)

5. Show that for the probability function 1

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

0, otherwise

xx xp x P X x

( )E X does not exist. (N/D 2012)

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

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

2j

P X j j Find

(1) Mean of X

(2) ( is even)P X and

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

Moments and Moment Generating Function

1. Find the MGF of the two parameter exponential distribution whose density function is

given by ( )( ) ,

x af x e x a

and hence find the mean and variance.

(A/M 2010)



2. Derive the m.g.f of Poisson distribution and hence or otherwise deduce its mean and

variance. (A/M 2011)

3. If the probability density of X is given by 2(1 ) for 0 1

( )0, otherwise

x xf x

, find its rth

moment. Hence evaluate 2

2 1E X

. (N/D 2012)

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

2 , 0( ) 4

0, elsewhere

xx

e xf x

. Also deduce the first four moments about the origin.

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

5. Find MGF corresponding to the distribution 2

1, 0

( ) 2

0, otherwise

ef

and hence find

its mean and variance. (N/D 2012)

Problems on distributions

1. 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? (A/M 2010)

2. 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)

3. 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 2of them will have marks

over 70? (A/M 2011)

4. 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) at most 35.0 cc/min

(3) anywhere from 30.0 to 40.0 cc/mm. (N/D 2012)

5. 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)

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

find the mean and standard deviation of the distribution. (M/J 2012)

7. If X and Y are independent random variables following (8, 2)N and 12,4 3N

respectively, find the value of such that 2 2 2P X Y P X Y .

(N/D 2010)

8. 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? (M/J 2012)

Function of random variable

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

sin2

XY

. (N/D 2010)

2. If X is a uniform random variable in the interval @, find the probability density function

Y X and E Y . (N/D 2011)

3. The random variable X has exponential distribution with, 0

( )0, otherwise

xe x

f x

.

Find the density function of the variable given by

(1) 3 5Y X

(2) 2Y X (N/D 2012)



Unit – II (Two Dimensional Random Variables form)

Joint distributions – Marginal & Conditional

1. The joint p.d.f of two dimensional random variable (X,Y) is given by 8

( , )9

f x y xy ,

0 2x y and ( , ) 0f x y , otherwise. Find the densities of X and Y, and the

conditional densities ( / )f x y and ( / )f y x . (A/M 2010)

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

8, 1 2

( , ) 9

0, otherwise

xyx y

f x y

. Find the conditional density functions of X and Y .

(N/D 2011)

3. The joint pdf of a two-dimensional random variable (X,Y) is given by 2

2( , ) ,

8

xf x y xy

0 2,0 1x y . Compute ( 1 / 2)P Y ,

( 1 / 1 / 2)P X Y and ( 1)P X Y . (N/D 2012)

4. 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. (A/M 2010)

Covariance, Correlation and Regression

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

2. The joint probability density function of random variable ,X Y is given by

2 2

( , ) , 0, 0x y

f x y Kxye x y

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

and Y independent? (M/J 2012)



3. The joint probability density function of the two dimensional random variable ,X Y

is2 , 0 1, 0 1

( , )0, otherwise

x y x yf x y

. Find the correlation coefficient

between X and Y . (N/D 2011)

4. Two random variables X and Y have the joint probability density function given by 2

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

0, otherwiseXY

k x y x yf x y

.

(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 .

(N/D 2010)

5. 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)

6. 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)

7. The regression equation of X on Y is 3 5 108 0Y 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. (A/M 2011)

Transformation of the random variables

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

1, 1 2( )

0, otherwiseX

xf x

and , 2 4

( ) 6

0, otherwiseY

yy

f y

, find the density function of

Z XY . (A/M 2011)

2. X and Y are independent with a common PDF (exponential): , 0

( )0, 0

xe x

f xx

and

, 0( )

0, 0

ye y

f yy

. Find the PDF for X Y . (N/D 2011)

3. If X and Y are independent random variables with probability density functions 4

( ) 4 , 0;x

Xf x e x

2( ) 2 , 0

y

Yf y e y

respectively.



(i) Find the density function of , X

U V X YX Y

(ii) Are U and V independent?

(iii) What is 0.5P U ?

4. Let ,X Y be a two dimensional random variable and the probability density function

be given by ( , ) , 0 , 1f x y x y x y . Find the p.d.f of U XY . (M/J 2012)

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

U X Y is given by ( ) ( ) ( )x y

h u f v f u v dv

. (N/D 2010)

Central Limit Theorem

1. 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. (A/M 2010)

2. 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.

(A/M 2011)

3. Let 1 2 3, , , ...

nX X X X be Poisson variates with parameter 2 and

1 2 3...

n nS X X X X where 75n . Find 120 160

np S using central

limit theorem. (M/J 2012)

4. If1 2 3, , , ...

nX X X X are uniform variates with mean 2.5 and variance 3 / 4 , use CLT

to estimate 108 12.6n

p S where 1 2 3

... , 48n n

S X X X X n .

(N/D 2011)

5. If , 1,2,3...20i

V i 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

iV is uniformly distributed over (0,10). (N/D 2010)



Unit – III (Classification of Random Processes)

Verification of SSS and WSS process

1. Examine whether the random process ( ) cos( )X t A t is a wide sense

stationary if A and are constants and is uniformly distributed random variable in

(0,2π). (A/M 2010),(N/D 2011)

2. A random process ( )X t defined by ( ) cos sin , X t A t B 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. (A/M 2011)

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

1

1

( ), 1,2...

(1 )( )

, 01

n

n

atn

atP X t n

atn

at

. Find the mean and variance of the process.

Is the process first-order stationary? (N/D 2010),(N/D 2011),(N/D 2012)

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

, determine the second

order moment of the RV (8) (5)X X . (M/J 2012)

Ergodic Processes, Mean ergodic and Correlation ergodic

1. The random binary transmission process ( )X t is a wide sense process with zero mean

and autocorrelation function ( ) 1RT

, 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?

(A/M 2010)

2. A random process has sample functions of the form ( ) cosX t A t , 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 2 . Assume that the random

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

(A/M 2011)



3. 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.

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

Problems on Markov Chain

1. The transition probability matrix of a Markov chain ( )X t , 1, 2, 3, ...n having three

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)0.7 0.2 0.1P , Find 2

3P X and 3 2 1 02, 3, 3, 2P X X X X .

(A/M 2010)

Poisson process

1. If the process ( ); 0X t t is a Poisson process with parameter , obtain

( )P X t n . Is the process first order stationary? (N/D 2010),(N/D 2012)

2. 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)

3. 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)

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

of duration t seconds, is a Poisson process with mean 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)

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

parameter has an exponential distribution with mean 1

. (A/M 2011)



Normal (Gaussian) & Random telegraph Process

1. If ( )X t is a Gaussian process with ( ) 10t and 1 2

1 2, 16

t tC t t e

, find the

probability that

(1) (10) 8X

(2) (10) (6) 4X X (A/M 2011)

2. Suppose that ( )X t is a Gaussian process with 2,x

0.25

xxR e

. Find the

probability that (4) 1X . (M/J 2012)

3. 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 value

1 and 1 with equal probability and 1 22

1 2( , )

t t

XXR t t e

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

Unit – IV (Correlation and Spectral densities)

Auto Correlation from the given process

1. Find the autocorrelation function of the periodic time function of the period time

function ( ) sinX t A t . (A/M 2010)

Relationship between XXR and XX

S

1. The autocorrelation function of the random binary transmission ( )X t is given by

( ) 1RT

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

process ( )X t . (A/M 2010)

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

is 1 , for 1

( )0, elsewhere

R

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

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

2

0cos

2XX

AR . (M/J 2012)



4. The autocorrelation function of a random process is given by

2

2

;

( )1 ;

R

. Find the power spectral density of the process.

(N/D 2011)

5. The Auto correlation function of a WSS process is given by 22

( )R e

determine

the power spectral density of the process. (A/M 2011)

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

0( ) 1 / ,

xxR A T T t T . (N/D 2012)

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

density is given by 2( ) 1

XXS for 1 and ( ) 0

XXS for 1 .(A/M 2010)

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

01,

( )0, otherwise

S

. Find ( )R and show that ( )X t and0

X t

are

uncorrelated. (A/M 2011)

Relationship between XYR and XY

S

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

0 0( , ) sin( ) cos 2

2XY

ABR t t t where ,A B and

0 are constants.

Find the cross-power spectrum ( )XY

S . (M/J 2012)

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

, for 1( )

0, elsewherexy

a bjS

. Find the cross correlation function.

(N/D 2010),(A/M 2011),(N/D 2011)



Properties, Theorem and Special problems

1. State and prove Weiner – Khintchine Theorem.

(N/D 2010),(A/M 2011),(N/D 2011),(N/D2012)

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

( )XX

R and ( )YY

R respectively then prove that ( ) (0) (0)XY XX YY

R R R .

Establish any two properties of auto correlation function ( )XX

R .(N/D 2010),(N/D2012)

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

4 2

9

5 4XX

S

.

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

10( ) 4XX

R e

. Find the mean and variance of 1

0

( ) Y X t dt . (M/J 2012)

5. ( )X t and ( )Y t are zero mean and stochastically independent random processes

having autocorrelation functions ( )XX

R e

and ( ) cos2YY

R 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)

6. 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.

(2) The power spectrum of ( )Z t if ( )X t and ( )Y t are orthogonal.

(M/J 2012)



Unit – V (Linear systems with Random inputs)

Input and Output process

1. 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)

2. 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 ( )XY

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

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

function ( )XY

R and the output autocorrelation function ( )YY

R . (N/D 2011)

4. Consider a system with transfer function 1

1 j. An input signal with autocorrelation

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

value of the output. (A/M 2011),(M/J 2012)

5. If ( )X t is a WSS process and if ( ) ( ) ( )Y t h X t d

then prove that

(1) ( ) ( )* ( )XY XX

R R h where * stands for convolution.

(2) *( ) ( ) ( )

XY XXS S H . (M/J 2012)

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

( ) 1H for 0 0

. If the autocorrelation function of the input process is

0 ( )2

Nt , find the autocorrelation function of the output process. (A/M 2010)

7. 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 0X

and 2

( )XX

R e

. Find the mean Y

and

power spectrum ( )YY

S of the output if the system transfer function is given by

1( )

2H

i

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



Input and Output process with impulse response

1. A system has an impulse response ( ) ( )t

h t e U t , find the power spectral density of

the output ( )Y t corresponding to the input ( )X t . (N/D 2010),(N/D 2012)

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

( ) ( )XX

R A

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

impulse function. The linear system has the impulse response of ( ) ( )bt

h t e u t where

( )u t represents the unit step function. Find ( )YY

R . Also find the mean and variance of

( )Y t . (A/M 2011),(M/J 2012)

3. A wide sense stationary random process ( )X t with autocorrelation ( )a

XXR e

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

transmission input system with impulse response ( ) ( )bt

h t e u t where b is a real

positive constant. Find the autocorrelation of the output ( )Y t of the system.(A/M 2010)

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

( ) ( )

t

RCh t e u tRC

. Assume an

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

correlation function of the output process. (A/M 2011)

5. 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 that2

( ) ( ) ( )yy xx

S H S .

(N/D 2011)

Band Limited White Noise

1. If 0

( ) cos( ) ( )Y t A t N t , where A is a constant, is a random variable with a

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

with power spectral density 0

0, for

( ) 2

0, elsewhere

B

NN

N

S

. Find the power

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

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



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

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

a power spectral density 0( )2

NN

NS for

0 B and ( ) 0

NNS , elsewhere.

Find the power spectral density of ( )Y t , assuming that ( )N t and are independent.

(A/M 2010)

3. If ( )N t is a band limited white noise centered at a carrier frequency 0

such that

0

0, for

( ) 2

0, elsewhere

B

NN

N

S

. Find the autocorrelation of ( )N t .

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

4. If ( )X t is a band limited process such that ( ) 0XX

S when , prove that

2 22 (0) ( ) (0)

XX XX XXR R R . (A/M 2010)

5. A white Gaussian noise ( )X t with zero mean and spectral density 0

2

Nis applied to a

low-pass RC filter shown in the figure.

Determine the autocorrelation of the output ( )Y t . (N/D 2011)

----All the Best----



SUBJECT NAME : Probability &Random Process


MATERIAL NAME : Part – A questions




1) If the p.d.f of a random variable X is ( )2

xf x in 0 2x , find 1.5 / 1P X X .

2) If the MGF of a uniform distribution for a random variable X is 5 41 t te e

t , find ( )E X .

3) The moment generating function of a random variable X is given by 3 1

( )t

e

M t e

. What

is 0P X ?

4) The CDF of a continuous random variable is given by /5

0, 0( )

1 , 0x

xF x

e x

. Find

the PDF and mean of X .

5) Establish the memoryless property of the exponential distribution.

6) Find C , if 2

; 1,2, ...3

n

P X n C n

.

7) 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?

8) 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.

9) A continuous random variable X has probability density function2

3 , 0 1( )

0, otherwise

x xf x

. Find k such that 0.5P X k .

10) If X is uniformly distributed in ,2 2

. Find the pdf of tanY X .

11) If X is a normal random variable with mean zero and variance 2 , find the PDF of XY e .



Unit – II (Two Dimensional Random Variables)

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

otherwise, is to be the joint density function.

2) A random variable X has mean 10 and variance 16. Find the lower bound for

(5 15)P X .

3) Let X and Y be continuous random variables with joint probability density function

( )( , ) , 0 2,

8XY

x x yf x y x x y x

and ( , ) 0

XYf x y elsewhere. Find

/( / )

Y Xf y x .

4) Find the marginal density functions of X and Y if

26, 0 1, 0 1

( , ) 5

0, otherwise

x y x yf x y

.

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

regression.

6) Let X and Y be two discrete random variables with joint probability mass function

12 , 1, 2 and 1,2

, 18

0, otherwise

x y x yP X x Y y

. Find the marginal probability

mass functions of X and Y .

7) State Central Limit Theorem for iid random variables.

8) If the joint pdf of ,X Y is

, 0, 0( , )

0, otherwise

x y

XY

e x yf x y

, check whether X and

Y are independent.

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

coefficient between X and Y .


1) Define a wide sense stationary process.

2) Define a strictly stationary random process.

3) Define a Markov chain and give an example.

4) Prove that a first order stationary process has a constant mean.

5) State the postulates of a Poisson process.

6) Prove that sum of two independent Poisson processes is again a Poisson process.



7) If ( )X t is a normal process with ( ) 10t and 1 2

1 2, 16

t tC t t e

find the variance of

(10) (6)X X .

8) Consider the random process ( ) cos( )X t t , where is a random variable with density

function 1

( ) , 2 2

f

. Check whether or not the process is wide sense

stationary.

9) When is a random process said to be mean ergodic?

Unit – IV (Correlation and Spectral densities) 1) Find the power spectral density function of the stationary process whose autocorrelation

function is given by e

.

2) The autocorrelation function of a stationary random process is2

9( ) 16

1 16R

. Find

the mean and variance of the process.

3) Prove that for a WSS process ( ) , ( , )XX

X t R t t is an even function of .

4) Prove that ( ) ( )xy yx

S S .

5) Find the variance of the stationary process ( )x t whose auto correlation function is given

by 2

( ) 2 4XX

R e

.

6) State any two properties of cross correlation function.


1) Define time – invariant system.

2) Define Band-Limited white noise.

3) State autocorrelation function of the white noise.

4) Find the system Transfer function, if a Linear Time Invariant system has an impulse function

1 ;

2( )

0 ;

t ccH t

t c

.

5) Define white noise.

6) Prove that the system ( ) ( ) ( )y t h u X t u du

is a linear time-invariant system.

7) What is unit impulse response of a system? Why is it called so?

8) 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 .

9) Sate any two properties of a linear time – invariant system.



10) 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 function related.

----All the Best----

Engineering Mathematics Material 2012

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 1



MATERIAL NAME : Formula Material



UNIT-I (RANDOM VARIABLES)

1) Discrete random variable: A random variable whose set of possible values is either finite or countably infinite is called discrete random variable. Eg: (i) Let X represent the sum of the numbers on the 2 dice, when two dice are thrown. In this case the random variable X takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. So X is a discrete random variable. (ii) Number of transmitted bits received in error.

2) Continuous random variable: A random variable X is said to be continuous if it takes all possible values between certain limits. Eg: The length of time during which a vacuum tube installed in a circuit functions is a continuous random variable, number of scratches on a surface, proportion of defective parts among 1000 tested, number of transmitted in error.

3) Sl.No. Discrete random variable Continuous random variable 1

( ) 1i

i

p x

( ) 1f x dx

2 ( )F x P X x ( ) ( )

x

F x P X x f x dx

3 Mean ( )i i

i

E X x p x Mean ( )E X xf x dx

4 2 2( )

i i

i

E X x p x 2 2( )E X x f x dx

5 2

2Var X E X E X

22

Var X E X E X

6 Moment = r r

i i

i

E X x p Moment = ( )

r rE X x f x dx

7 M.G.F M.G.F



( )tX tx

X

x

M t E e e p x ( )

tX tx

XM t E e e f x dx

4) E aX b aE X b

5) 2Var VaraX b a X

6) 2 2Var VaraX bY a X b Var Y

7) Standard Deviation Var X

8) ( ) ( )f x F x 9) ( ) 1 ( )p X a p X a

10)

/p A B

p A Bp B

, 0p B

11) If A and B are independent, then p A B p A p B .

12) 1st Moment about origin = E X = 0

Xt

M t

(Mean)

2nd Moment about origin = 2E X =

0X

t

M t

The co-efficient of !

rt

r = r

E X (rth Moment about the origin)

13) Limitation of M.G.F: i) A random variable X may have no moments although its m.g.f exists.

ii) A random variable X can have its m.g.f and some or all moments, yet the m.g.f does not generate the moments.

iii) A random variable X can have all or some moments, but m.g.f does not exist except perhaps at one point.

14) Properties of M.G.F: i) If Y = aX + b, then bt

Y XM t e M at .

ii) cX XM t M ct , where c is constant.

iii) If X and Y are two independent random variables then

X Y X YM t M t M t

.

15) P.D.F, M.G.F, Mean and Variance of all the distributions: Sl.No.

Distribution

P.D.F ( ( )P X x ) M.G.F Mean Variance

1 Binomial x n x

xnc p q

n

tq pe

np npq

2 Poisson

!

xe

x

1

te

e

3 Geometric 1xq p

(or) xq p

1

t

t

pe

qe

1

p

2

q

p



4 Uniform 1

, ( )

0, otherwise

a x bf x b a

( )

bt ate e

b a t

2

a b

2( )

12

b a

5 Exponential

, 0, 0( )

0, otherwise

xe xf x

t

1

2

1

6 Gamma 1

( ) , 0 , 0( )

xe x

f x x

1

(1 )t

7 Normal 21

21( )

2

x

f x e

2 2

2

tt

e

2

16) Memoryless property of exponential distribution

/P X S t X S P X t .

17) Function of random variable: ( ) ( )Y X

dxf y f x

dy

UNIT-II (RANDOM VARIABLES)

1) 1ij

i j

p (Discrete random variable)

( , ) 1f x y dxdy

(Continuous random variable)

2) Conditional probability function X given Y ,

/( )

i i

P x yP X x Y y

P y .

Conditional probability function Y given X ,

/( )

i i

P x yP Y y X x

P x .

,

/( )

P X a Y bP X a Y b

P Y b

3) Conditional density function of X given Y, ( , )

( / )( )

f x yf x y

f y .

Conditional density function of Y given X, ( , )

( / )( )

f x yf y x

f x .

4) If X and Y are independent random variables then

( , ) ( ). ( )f x y f x f y (for continuous random variable)



, .P X x Y y P X x P Y y (for discrete random variable)

5) Joint probability density function , ( , )

d b

c a

P a X b c Y d f x y dxdy .

0 0

, ( , )

b a

P X a Y b f x y dxdy

6) Marginal density function of X, ( ) ( ) ( , )X

f x f x f x y dy

Marginal density function of Y, ( ) ( ) ( , )Y

f y f y f x y dx

7) ( 1) 1 ( 1)P X Y P X Y

8) Correlation co – efficient (Discrete): ( , )

( , )X Y

Cov X Yx y

1( , )Cov X Y XY XY

n , 2 21

XX X

n , 2 21

YY Y

n

9) Correlation co – efficient (Continuous): ( , )

( , )X Y

Cov X Yx y

( , ) ,Cov X Y E X Y E X E Y , ( )X

Var X , ( )Y

Var Y

10) If X and Y are uncorrelated random variables, then ( , ) 0Cov X Y .

11) ( )E X xf x dx

, ( )E Y yf y dy

, , ( , )E X Y xyf x y dxdy

.

12) Regression for Discrete random variable:

Regression line X on Y is xyx x b y y ,

2xy

x x y yb

y y

Regression line Y on X is yxy y b x x ,

2yx

x x y yb

x x

Correlation through the regression, .XY YX

b b Note: ( , ) ( , )x y r x y



13) Regression for Continuous random variable:

Regression line X on Y is ( ) ( )xy

x E x b y E y , x

xy

y

b r

Regression line Y on X is ( ) ( )yx

y E y b x E x , y

yx

x

b r

Regression curve X on Y is / / x E x y x f x y dx

Regression curve Y on X is / / y E y x y f y x dy

14) Transformation Random Variables:

( ) ( )Y X

dxf y f x

dy (One dimensional random variable)

( , ) ( , )UV XY

u u

x yf u v f x y

v v

x y

(Two dimensional random variable)

15) Central limit theorem (Liapounoff’s form)

If X1, X2, …Xn be a sequence of independent R.Vs with E[Xi] = µi and Var(Xi) = σi2, i

= 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain general conditions, Sn

follows a normal distribution with mean 1

n

i

i

and variance 2 2

1

n

i

i

as

n .

16) Central limit theorem (Lindberg – Levy’s form)

If X1, X2, …Xn be a sequence of independent identically distributed R.Vs with E[Xi]

= µi and Var(Xi) = σi2, i = 1,2,…n and if Sn = X1 + X2 + … + Xn then under certain

general conditions, Sn follows a normal distribution with mean n and variance

2n as n .

Note: nS nz

n

( for n variables),

Xz

n

( for single variables)



UNIT-III (MARKOV PROCESSES AND MARKOV CHAINS)

1) Random Process:

A random process is a collection of random variables {X(s,t)} that are

functions of a real variable, namely time ‘t’ where s Є S and t Є T.

2) Classification of Random Processes:

We can classify the random process according to the characteristics of time t

and the random variable X. We shall consider only four cases based on t and X

having values in the ranges -∞< t <∞ and -∞ < x < ∞.

Continuous random process

Continuous random sequence

Discrete random process

Discrete random sequence

Continuous random process:

If X and t are continuous, then we call X(t) , a Continuous Random Process.

Example: If X(t) represents the maximum temperature at a place in the

interval (0,t), {X(t)} is a Continuous Random Process.

Continuous Random Sequence:

A random process for which X is continuous but time takes only discrete values is

called a Continuous Random Sequence.

Example: If Xn represents the temperature at the end of the nth hour of a day, then

{Xn, 1≤n≤24} is a Continuous Random Sequence.

Discrete Random Process:

If X assumes only discrete values and t is continuous, then we call such random

process {X(t)} as Discrete Random Process.

Example: If X(t) represents the number of telephone calls received in the interval

(0,t) the {X(t)} is a discrete random process since S = {0,1,2,3, . . . }

Discrete Random Sequence:

A random process in which both the random variable and time are discrete is called

Discrete Random Sequence.

Example: If Xn represents the outcome of the nth toss of a fair die, the {Xn : n≥1} is a

discrete random sequence. Since T = {1,2,3, . . . } and S = {1,2,3,4,5,6}



3) Condition for Stationary Process: ( ) ConstantE X t , ( ) constantVar X t .

If the process is not stationary then it is called evolutionary.

4) Wide Sense Stationary (or) Weak Sense Stationary (or) Covariance Stationary: A random process is said to be WSS or Covariance Stationary if it satisfies the

following conditions.

i) The mean of the process is constant (i.e) ( ) constantE X t .

ii) Auto correlation function depends only on (i.e)

( ) ( ). ( )XX

R E X t X t

5) Time average:

The time average of a random process ( )X t is defined as1

( ) 2

T

T

T

X X t dtT

.

If the interval is 0,T , then the time average is 0

1( )

T

TX X t dt

T .

6) Ergodic Process:

A random process ( )X t is called ergodic if all its ensemble averages are

interchangeable with the corresponding time average TX .

7) Mean ergodic:

Let ( )X t be a random process with mean ( )E X t and time average TX ,

then ( )X t is said to be mean ergodic if TX as T (i.e)

( )T

TE X t Lt X

.

Note: var 0T

TLt X

(by mean ergodic theorem)

8) Correlation ergodic process:

The stationary process ( )X t is said to be correlation ergodic if the process

( )Y t is mean ergodic where ( ) ( ) ( )Y t X t X t . (i.e) ( )T

TE Y t Lt Y

.

Where TY is the time average of ( )Y t .

9) Auto covariance function:

( ) ( ) ( ) ( )XX XX

C R E X t E X t

10) Mean and variance of time average:

Mean: 0

1( )

T

TE X E X t dt

T

Variance: 2

2

1( ) ( )

2

T

T XX XX

T

Var X R C dT



11) Markov process: A random process in which the future value depends only on the present value and not on the past values, is called a markov process. It is symbolically

represented by 1 1 1 1 0 0

( ) / ( ) , ( ) ... ( )n n n n n n

P X t x X t x X t x X t x

1 1( ) / ( )

n n n nP X t x X t x

Where 0 1 2 1...

n nt t t t t

12) Markov Chain:

If for all n , 1 1 2 2 0 0

/ , , ...n n n n n n

P X a X a X a X a

1 1/

n n n nP X a X a

then the process n

X , 0,1,2, ...n is called the

markov chain. Where 0 1 2, , , ... , ...

na a a a are called the states of the markov chain.

13) Transition Probability Matrix (tpm): When the Markov Chain is homogenous, the one step transition probability is denoted by Pij. The matrix P = {Pij} is called transition probability matrix.

14) Chapman – Kolmogorov theorem: If ‘P’ is the tpm of a homogeneous Markov chain, then the n – step tpm P(n) is

equal to Pn. (i.e) ( )n

n

ij ijP P .

15) Markov Chain property: If 1 2 3, , , then P and

1 2 31 .

16) Poisson process: If ( )X t represents the number of occurrences of a certain event in (0, )t ,then

the discrete random process ( )X t is called the Poisson process, provided the

following postulates are satisfied.

(i) 1 occurrence in ( , )P t t t t O t

(ii) 0 occurrence in ( , ) 1P t t t t O t

(iii) 2 or more occurrences in ( , )P t t t O t

(iv) ( )X t is independent of the number of occurrences of the event in any

interval.

17) Probability law of Poisson process:

( ) , 0,1,2, ...!

xte t

P X t x xx

Mean ( )E X t t , 2 2 2( )E X t t t , ( )Var X t t .

UNIT-IV (CORRELATION AND SPECTRAL DENSITY)

XXR - Auto correlation function

XXS - Power spectral density (or) Spectral density



XYR - Cross correlation function

XYS - Cross power spectral density

1) Auto correlation to Power spectral density (spectral density):

i

XX XXS R e d

2) Power spectral density to Auto correlation:

1

2

i

XX XXR S e d

3) Condition for ( )X t and ( )X t are uncorrelated random process is

( ) ( ) ( ) ( ) 0XX XX

C R E X t E X t

4) Cross power spectrum to Cross correlation:

1

2

i

XY XYR S e d

5) General formula:

i) 2 2cos cos sin

axax e

e bx dx a bx b bxa b

ii) 2 2sin sin cos

axax e

e bx dx a bx b bxa b

iii) 2 2

2

2 4

a ax ax x

iv) sin2

i ie e

i

v) cos2

i ie e

UNIT-V (LINEAR SYSTEMS WITH RANDOM INPUTS)



1) Linear system:

f is called a linear system if it satisfies

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )f a X t a X t a f X t a f X t

2) Time – invariant system:

Let ( ) ( )Y t f X t . If ( ) ( )Y t h f X t h then f is called a time –

invariant system.

3) Relation between input ( )X t and output ( )Y t :

( ) ( ) ( ) Y t h u X t u du

Where ( )h u system weighting function.

4) Relation between power spectrum of ( )X t and output ( )Y t :

2( ) ( ) ( )

YY XXS S H

If ( )H is not given use the following formula ( ) ( ) j t

H e h t dt

5) Contour integral:

2 2

imxm ae

ea x a

(One of the result)

6) 1

2 2

1

2

ae

Fa a

(from the Fourier transform)

---- All the Best ----





MATERIAL NAME : Problem Material




Problems on Discrete & Continuous R.Vs 1) A random variable X has the following probability function:

X 0 1 2 3 4 5 6 7

P(X) 0 K 2K 2K 3K K2 2K2 7K2 + K

a) Find K .

b) Evaluate 6 , 6P X P X .

c) Find 2 , 3 , 1 5P X P X P X .

d) If 1

2P X C , find the minimum value of C .

e) 1.5 4.5 / 2P X X

2) The probability function of an infinite discrete distribution is given by

1

, 1,2,3...2

jP X j j . Find the mean and variance of the distribution.

Also find X is evenP , 5P X and X is divisible by 3P .

3) Suppose that X is a continuous random variable whose probability density function is

given by 24 2 , 0 2

( )0, otherwise

C x x xf x

(a) find C (b) find 1P X .

4) A continuous random variable X has the density function

2( ) ,

1

Kf x x

x

. Find the value of K ,the distribution function and

0P X .



5) A random variable X has the p.d.f 2 , 0 1

( )0, otherwise

x xf x

. Find (i) 1

2P X

(ii)

1 3

2 4P X

(iii)3 1

/4 2

P X X

(iv) 3 1

/4 2

P X X

.

6) If a random variable X has the p.d.f

1, 2

( ) 4

0, otherwise

xf x

. Find (a) 1P X

(b) 1P X (c) 2 3 5P X

7) The amount of time, in hours that a computer functions before breaking down is a

continuous random variable with probability density function given by

100 , 0( )

0, 0

x

e xf x

x

. What is the probability that (a) a computer will function

between 50 and 150 hrs. before breaking down (b) it will function less than 500 hrs.

8) A random variable X has the probability density function

, 0( )

0, otherwise

xxe xf x

. Find , . . , 2 5 , 7c d f P X P X .

9) If the random variable X takes the values 1,2,3 and 4 such that

2 1 3 2 3 5 4P X P X P X P X . Find the probability

distribution.

10) The distribution function of a random variable X is given by

( ) 1 1 ; 0x

F x x e x . Find the density function, mean and variance of X.

11) A continuous random variable X has the distribution function

4

0, 1

( ) ( 1) , 1 3

0, 30

x

F x k x x

x

. Find k , probability density function ( )f x , 2P X .

12) A test engineer discovered that the cumulative distribution function of the lifetime

of an equipment in years is given by51 , 0( )

0, 0

x

e xF x

x

.

i) What is the expected life time of the equipment?

ii) What is the variance of the life time of the equipment?

Moments and Moment Generating Function



1) Find the moment generating function of R.V X whose probability function

1( ) , 1,2, ...

2x

P X x x Hence find its mean and variance.

2) The density function of random variable X is given by ( ) (2 ), 0 2f x Kx x x .

Find K, mean, variance and rth moment.

3) Let X be a R.V. with p.d.f3

1, 0

( ) 3

0, Otherwise

x

e xf x

. Find the following

a) P(X > 3).

b) Moment generating function of X.

c) E(X) and Var(X).

4) Find the MGF of a R.V. X having the density function, 0 2

( ) 2

0, otherwise

xx

f x

. Using

the generating function find the first four moments about the origin.

5) Define Binomial distribution and find the M.G.F, Mean and Variance of the Binomial

distribution.

6) Define Poisson distribution and find the M.G.F, Mean and Variance of the Poisson

distribution.

7) Define Geometric distribution and find the M.G.F, Mean and Variance of the

Geometric distribution.

8) Write the pdf of Uniform distribution and find the M.G.F, Mean and Variance.

9) Define Exponential distribution and find the M.G.F, Mean and Variance of the

Exponential distribution.

10) Define Gamma distribution and find the M.G.F, Mean and Variance of the Gamma

distribution.

11) Define Normal distribution and find the M.G.F, Mean and Variance of the Normal

distribution.

Problems on distributions

1) The mean of a Binomial distribution is 20 and standard deviation is 4. Determine the

parameters of the distribution.

2) If 10% of the screws produced by an automatic machine are defective, find the

probability that of 20 screws selected at random, there are (i) exactly two defectives

(ii) atmost three defectives (iii) atleast two defectives and (iv) between one and

three defectives (inclusive).

3) In a certain factory furning razar blades there is a small chance of 1/500 for any

blade to be defective. The blades are in packets of 10. Use Poisson distribution to



calculate the approximate number of packets containing (i) no defective (ii) one

defective (iii) two defective blades respectively in a consignment of 10,000 packets.

4) The number of monthly breakdown of a computer is a random variable having a

Poisson distribution with mean equally to 1.8. Find the probability that this

computer will function for a month

a) Without a breakdown

b) With only one breakdown and

c) With atleast one breakdown.

5) Prove that the Poisson distribution is a limiting case of binomial distribution.

6) If the mgf of a random variable X is of the form 8(0.4 0.6)

te , what is the mgf of

3 2X . Evaluate E X .

7) A discrete R.V. X has moment generating function 5

1 3( )

4 4

t

XM t e

. Find

E X , Var X and 2P X .

8) If X is a binomially distributed R.V. with ( ) 2E X and 4

( )3

Var X , find 5P X .

9) If X is a Poisson variate such that 2 9 4 90 6P X P X P X , find the

mean and variance.

10) The number of personal computer (PC) sold daily at a CompuWorld is uniformly

distributed with a minimum of 2000 PC and a maximum of 5000 PC. Find the

following

(i) The probability that daily sales will fall between 2,500 PC and 3,000 PC.

(ii) What is the probability that the CompuWorld will sell at least 4,000 PC’s?

(iii) What is the probability that the CompuWorld will exactly sell 2,500 PC’s?

11) Suppose that a trainee soldier shoots a target in an independent fashion. If the

probability that the target is shot on any one shot is 0.8. (i) What is the probability

that the target would be hit on 6th attempt? (ii) What is the probability that it takes

him less than 5 shots? (iii) What is the probability that it takes him an even number

of shots?

12) A die is cast until 6 appears. What is the probability that it must be cast more than 5

times?

13) The length of time (in minutes) that a certain lady speaks on the telephone is found

to be random phenomenon, with a probability function specified by the function.

5 , 0( )

0, otherwise

x

Ae xf x

. (i) Find the value of A that makes f(x) a probability



density function. (ii) What is the probability that the number of minutes that she will

talk over the phone is (a) more than 10 minutes (b) less than 5 minutes and (c)

between 5 and 10 minutes.

14) If the number of kilometers that a car can run before its battery wears out is

exponentially distributed with an average value of 10,000 km and if the owner

desires to take a 5000 km trip, what is the probability that he will be able to

complete his trip without having to replace the car battery? Assume that the car has

been used for same time.

15) The mileage which car owners get with a certain kind of radial tyre is a random

variable having an exponential distribution with mean 40,000 km. Find the

probabilities that one of these tyres will last (i) atleast 20,000 km and (ii) atmost

30,000 km.

16) If a continuous random variable X follows uniform distribution in the interval 0,2

and a continuous random variable Y follows exponential distribution with

parameter , find such that 1 1P X P Y .

17) If X is exponantially distributed with parameter , find the value of K there exists

P X ka

P X k

.

18) State and prove memoryless property of Geometric distribution.

19) State and prove memoryless property of Exponential distribution.

20) The time required to repair a machine is exponentially distributed with parameter ½.

What is the probability that the repair times exceeds 2 hours and also find what is

the conditional probability that a repair takes at least 10 hours given that its

duration exceeds 9 hours?

21) The weekly wages of 1000 workmen are normall distributed around a mean of Rs. 70

with a S.D. of Rs. 5. Estimate the number of workers whose weekly wages will be (i)

between Rs. 69 and Rs. 72, (ii) less than Rs. 69 and (iii) more than Rs. 72.

22) In a test on 2000 electric bulbs, it was found that the life of a particular make, was

normally distributed with an average life of 2040 hours and S.D. of 60 hours.

Estimate the number of bulbs lilkely to burn for (i) more than 2150 hours, (ii) less

than 1950 hours and (iii) more than 1920 hours but less than 2160 hours.

Function of random variable

1) Let X be a continuous random variable with p.d.f, 1 5

( ) 12

0, otherwise

xx

f x

, find the

probability density function of 2X – 3.



2) If X is a uniformly distributed RV in ,2 2

, find the pdf of tanY X .

3) If X has an exponential distribution with parameter 1, find the pdf of Y X .

4) If X is uniformly distributed in 1,1 , find the pdf of sin2

XY

.

5) If the pdf of X is ( ) , 0x

f x e x , find the pdf of 2

Y X .

6) If X is uniformly distributed in 0,1 find the pdf of 1

2 1Y

X

.

Unit – II (Two Dimensional Random Variables)

Joint distributions – Marginal & Conditional 1) The two dimensional random variable (X,Y) has the joint density function

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

27

x yf x y x y

. Find the marginal distribution of X and Y

and the conditional distribution of Y given X = x. Also find the conditional

distribution of X given Y = 1.

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

( , ) 2 3 , 0,1,2; 1,2,3P x y K x y x y . Find all the marginal and conditional

probability distributions. Also find the probability distribution of X Y and

3P X Y .

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

(6 ) ,0 2, 2 4( , )

0 ,otherwise

K x y x yf x y

. Find the following (i) the value of K;

(ii) 1, 3P x y ; (iii) 3P x y ; (iv) 1/ 3P x y

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

2 ,0 1, 0 2( , ) 3

0 ,otherwise

xyx x y

f x y

. Find (i) 1

2P X

; (ii) P Y X ; (iii)

1 1/

2 2P Y X

. Check whether the conditional density functions are valid.

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

2 2

( , ) , 0 ,x y

f x y Kxye x y

. Find the value of K and Prove that X and Y

are independent.



6) If the joint distribution function of X and Y is given by

( , ) 1 1 , 0, 0x y

F x y e e x y and "0" otherwise . (i) Are X and Y

independent? (ii) Find 1 3, 1 2P X Y .

Covariance, Correlation and Regression 1) Define correlation and explain varies type with example.

2) Find the coefficient of correlation between industrial production and export using

the following data:

Production (X) 55 56 58 59 60 60 62

Export (Y) 35 38 37 39 44 43 44

3) Let X and Y be discrete random variables with probability function

( , ) , 1,2,3; 1,221

x yf x y x y

. Find (i) ,Cov X Y (ii) Correlation co –

efficient.

4) Two random variables X and Y have the following joint probability density function.

2 , 0 1, 0 1( , )

0, otherwise

x y x yf x y

. Find Var X , Var Y and the

covariance between X and Y. Also find Correlation between X and Y. ( ( , )X Y ).

5) Let X and Y be random variables having joint density function.

2 23, 0 , 1

( , ) 2

0, otherwise

x y x yf x y

. Find the correlation coefficient ( , )X Y .

6) The independent variables X and Y have the probability density functions given by

4 , 0 1( )

0, otherwiseX

ax xf x

4 , 0 1

( )0, otherwise

Y

by yf y

. Find the correlation

coefficient between X and Y .

(or)

The independent variables X and Y have the probability density functions given by

4 , 0 1( )

0, otherwiseX

ax xf x

4 , 0 1

( )0, otherwise

Y

by yf y

. Find the correlation

coefficient between X Y and X Y .



7) Let X,Y and Z be uncorrelated random variables with zero means and standard

deviations 5, 12 and 9 respectively. If U X Y and V Y Z , find the

correlation coefficient between U and V .

8) 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 .

9) From the data, find

(i) The two regression equations.

(ii) The coefficient of correlation between the marks in Economics and

Statistics.

(iii) The most likely marks in statistics when a mark in Economics is 30.

Marks in Economics 25 28 35 32 31 36 29 38 34 32

Marks in Statistics 43 46 49 41 36 32 31 30 33 39

10) The two lines of regression are 8x – 10y + 66 = 0, 40x – 18y – 214 = 0. The variance

of X is 9. Find (i) the mean values of X and Y (ii) correlation coefficient between X

and Y (iii) Variance of Y .

11) The joint p.d.f of a two dimensional random variable is given by

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

3f x y x y x y . Find the following

(i) The correlation co – efficient.

(ii) The equation of the two lines of regression

(iii) The two regression curves for mean

Transformation of the random variables

1) If X is a uniformly distributed RV in ,2 2

, find the pdf of tanY X .

2) Let (X,Y) be a two – dimensional non – negative continuous random variables having

the joint probability density function 2 2

4 , 0, 0( , )

0, elsewhere

x yxye x y

f x y

. Find the

density function of 2 2U X Y .

3) X and Y be independent exponential R.Vs. with parameter 1. Find the j.p.d.f of

U X Y andX

VX Y

.

(Or) (The above problem may be ask as follows)



The waiting times X and Y of two customers entering a bank at different times are

assumed to be independent random variables with respective probability density

functions. , 0

( )0, otherwise

xe xf x

and , 0

( )0, otherwise

ye yf y

Find the joined p.d.f of the sum of their waiting times, U X Y and the fraction of

this time that the first customer spreads waiting, i.e X

VX Y

. Find the marginal

p.d.f’s of U and V and show that they are independent.

(Or)

If X and Y are independent random variable with pdf , 0x

e x and , 0

ye y , find the

density function of X

UX Y

and V X Y . Are they independent?

4) If X and Y are independent exponential random variables each with parameter 1,

find the pdf of U = X – Y.

5) Let X and Y be independent random variables both uniformly distributed on (0,1).

Calculate the probability density of X + Y.

6) Let X and Y are positive independent random variable with the identical probability

density function ( ) , 0x

f x e x . Find the joint probability density function of

U X Y andX

VY

. Are U and V independent?

7) If the joint probability density of X1and X2 is given by

1 2

1 21 2

, 0, 0( , )

0, elsewhere

x xe x x

f x x

, find the probability of 1

2 2

XY

X X

.

8) If X is any continuous R.V. having the p.d.f2 , 0 1

( )0, otherwise

x xf x

, and XY e

, find

the p.d.f of the R.V. Y.

9) If the joint p.d.f of the R.Vs X and Y is given by 2, 0 1

( , )0, otherwise

x yf x y

find the

p.d.f of the R.V. X

UY

.

10) Let X be a continuous random variable with p.d.f, 1 5

( ) 12

0, otherwise

xx

f x

, find the

probability density function of 2X – 3.



Central Limit Theorem

1) If 1 2, , ...

nX X X are Poisson variables with parameter 2 , use the Central Limit

Theorem to estimate (120 160)nP S where 1 2 ...n nS X X X and

75n .

2) The resistors 1 2 3 4, , and r r r r are independent random variables and is uniform in

the interval (450 , 550). Using the central limit theorem, find

1 2 3 4(1900 2100)P r r r r .

3) Let 1 2 100, ,...X X X be independent identically distributed random variables with

2 and2 1

4 . Find 1 2 100(192 ... 210)P X X X .

4) Suppose that orders at a restaurant are iid random variables with mean .8Rs

and standard deviation .2Rs . Estimate (i) the probability that first 100

customers spend a total of more than Rs.840 (ii) 1 2 100(780 ... 820)P X X X .

5) The life time of a certain brand of a Tube light may be considered as a random

variable with mean 1200 h and standard deviation 250 h. Find the probability, using

central limit theorem, that the average life time of 60 light exceeds 1250 h.

6) A random sample of size 100 is taken from a population whose mean is 60 and

variance is 400. Using Central limit theorem, with what probability can we assert

that the mean of the sample will not differ from 60 by more than 4.

7) A distribution with unknown mean has variance equal to 1.5. Use central limit

theorem to determine how large a sample should be taken from the distribution in

order that the probability will be at least 0.95 that the sample mean will be within

0.5 of the population mean.


Verification of SSS and WSS process 1) Define the following:

a) Markov process.

b) Independent increment random process.

c) Strict – sense stationary process.

d) Second order stationary process.



2) Classify the random process and give example to each.

3) Let cos( ) sin( )n

X A n B n where A and B are uncorrelated random variables

with 0E A E B and 1Var A Var B . Show that nX is covariance

stationary.

4) A stochastic process is described by ( ) sin cosX t A t B t where A and B are

independent random variables with zero means and equal standard deviations show

that the process is stationary of the second order.

5) If ( ) cos sinX t Y t Z t , where Y and Z are two independent random variables

with 2 2 2( ) ( ) 0, ( ) ( )E Y E Z E Y E Z and is a constants. Prove that

( )X t is a strict sense stationary process of order 2 (WSS).

6) At the receiver of an AM radio, the received signal contains a cosine carrier signal at

the carrier frequency 0 with a random phase that is uniformly distributed over

0,2 . The received carrier signal is 0( ) cosX t A t . Show that the

process is second order stationary.

7) The process ( ) :X t t T whose probability distribution, under certain conditions,

is given by

1

1

( ), 1, 2...

1( )

, 01

n

n

atn

atP X t n

atn

at

. Show that it is not stationary .

Ergodic Processes, Mean ergodic and Correlation ergodic

1) Consider the process ( ) cos sinX t A t B t where A and B are random variables

with ( ) ( ) 0E A E B and ( ) 0E AB . Prove that ( )X t is mean ergodic.

2) Prove that the random processes ( ) cosX t A t where A and are

constants and is uniformly distributed random variable in 0,2 is correlation

ergodic.

3) Consider the random process ( )X t with 2( ) cosX t A A t , where is a

uniformly distributed random variable in , . Prove that ( )X t is correlation

ergodic.

Note: The same problem they may ask by putting 10A .



4) Let ( )X t be a WSS process with zero mean and auto correlation function

( ) 1XX

RT

, 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?

Note: The same problem they may ask by putting 1T .

5) Given that the autocorrelation function for a stationary ergodic process with no

periodic components is 2

4( ) 25

1 6XX

R

. Find the mean and variance of the

process ( )X t .

Problems on Markov Chain

6) Consider a Markov chain ; 1n

X n with state space 1,2S and one – step

transition probability matrix0.9 0.1

0.2 0.8P

.

i) Is chain irreducible?

ii) Find the mean recurrence time of states ‘1’ and ‘2’.

iii) Find the invariant probabilities.

7) A raining process is considered as two state Markov chain. If it rains, it is considered

to be state 0 and if it does not rain, the chain is in state 1. The transitions probability

of the Markov chain is defined as0.6 0.4

0.2 0.8P

. Find the probability that it will

rain for 3 days. Assume the initial probabilities of state 0 and state 1 as 0.4 and 0.6

respectively.

8) A person owning a scooter has the option to switch over to scooter, bike or a car

next time with the probability of (0.3, 0.5, 0.2). If the transition probability matrix is

0.4 0.3 0.3

0.2 0.5 0.3

0.25 0.25 0.5

. What are the probabilities vehicles related to his fourth

purchase?

9) Assume that a computer system is in any one of the three states: busy, idle and

under repair respectively denoted by 0, 1, 2. Observing its state at 2 pm each day,

we get the transition probability matrix as

0.6 0.2 0.2

0.1 0.8 0.1

0.6 0 0.4

P

. Find out the 3rd

step transition probability matrix. Determine the limiting probabilities.



10) Two boys 1B and 2

B and two girls 1G and 2

G are throwing a ball from one to the

other. Each boys throws the ball to the other boy with probability 1/2 and to each

girl with probability 1/4. On the other hand each girl throws the ball to each boy

with probability 1/2 and never to the other girl. In the long run, how often does each

receive the ball?

11) A housewife buys 3 kinds of cereals A, B, C. She never buys the same cereal in

successive weeks. If she buys cereal A, the next week she buys cereal B. However if

she buys B or C the next week she is 3 times as likely to buy A as the other cereal.

How often she buys each of the 3 cereals?

12) Three boys A, B, C are throwing a ball each other. A always throws the 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. Find

the transition matrix and classify the states.

13) The transition probability matrix of a Markov chain 1,2,3...n n

X

having 3 states 1, 2

and 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)0.7,0.2,0.1P . Find

23P X and 3 2 1 0

2, 3, 3, 2P X X X X .

14) The tpm of a Markov chain with three states 0, 1, 2 is

3 / 4 1 / 4 0

1 / 4 1 / 2 1 / 4

0 3 / 4 1 / 4

P

and

the initial state distribution of the chain is 01/ 3, 0,1,2P X i i . Find (i)

22P X and (ii) 3 2 1 0

1, 2, 1, 2P X X X X .

Poisson process

1) Define Poisson process and obtain its probability distribution.

2) Prove that the Poisson process is Covariance stationary.

3) Show that the sum of two independent Poisson process is a Poisson process.

4) 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 mins.

(i) Exactly 4 customers arrive and

(ii) More than 4 customers arrive.

5) If customers arrive at a counter in accordance with a Poisson process with a mean

rate of 3 per minute, find the probability that the interval between 2 consecutive

arrivals is

(i) more than 1 minute



(ii) between 1 minute and 2 minutes

(iii) 4 minutes or less

6) A radar emits particles at the rate of 5 per minute according to Poisson distribution.

Each particles emitted has probability 0.6. Find the probability that 10 particles are

emitted in a 4 minutes period.

7) Queries presented in a computer data base are following a Poisson process of rate

6 queries per minute. An experiment consists of monitoring the data base for

m minutes and recording ( )N m the number of queries presented

i) What is the probability that no queries in a one minute interval?

ii) What is the probability that exactly 6 queries arriving in one minute

interval?

iii) What is the probability of less than 3 queries arriving in a half minute

interval?

Normal (Gaussian) & Random telegraph Process

1) Let ( )X t is a Gaussian random process with ( ) 10X t and

1 2

1 2( , ) 16

t t

XXC t t e

. Find the probability that (i) (10) 8X (ii) (10) (6) 4X X .

2) 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 values 1 and 1 with equal probability and 1 22 ( )

1 2( , )

t t

XXR t t e

.

Unit – IV (Correlation and Spectral densities)

Section – I 1) Determine the mean and variance of process given that the auto correlation

function 2

425

1 6XX

R

.

2) A stationary random process has an auto correlation function and is given by

2

2

25 36

6.25 4XX

R

. Find the mean and variance of the process.

3) If ( )X t and ( )Y t are two random processes then ( ) (0) (0)XY XX YY

R R R

where ( )XX

R and ( )YY

R are their respective auto correlation function.

4) If ( )X t and ( )Y t are two random processes then 1

( ) (0) (0)2

XY XX YYR R R

where ( )XX

R and ( )YY

R are their respective auto correlation function.

Section – II



5) State and Prove Wiener – Khinchine theorem.

6) The auto correlation of a stationary random process is given by

( ) , 0b

XXR ae b

. Find the spectral density function.

7) The auto correlation of the random binary transmission is given by

1 , ( )

0,

XX

for TR T

for T

. Find the power spectrum.

Note: By putting T = 1, the above problem can be ask1 , 1

( )0, 1

XX

forR

for

.

8) Show that the power spectrum of the auto correlation function 1e

is

3

22 2

4

.

9) Find the power spectral density of a WSS process with auto correlation function 2

( ) , 0XX

R e .

10) Find the power spectral density of the random process, if its auto correlation

function is given by ( ) cosXX

R e

.

11) Find the power spectral density function whose auto correlation function is given by 2

0( ) cos( )

2XX

AR .

Section – III

12) If the power spectral density of a WSS process is given by

, ( )

0, XX

ba a

aS

a

, find the auto correlation function of the process.

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

1, ( )

0, elsewhereXX

aS

. Find ( )XX

R and show that ( )X t and X ta

are

uncorrelated.

14) Find the autocorrelation function of the process ( )X t , for which the spectral

density is given by

21 , 1

( )0, 1

S

.



15) The cross – power spectrum of real random processes ( )X t and ( )Y t is given by

, 1( )

0, elsewhereXY

a jbS

. Find the cross – correlation function.

Section – IV

16) If ( ) ( ) ( )Y t X t a X t a ,prove that

( ) 2 ( ) ( 2 ) ( 2 )YY XX XX XX

R R R a R t a Hence prove that

2( ) 4sin ( ) ( )

YY XXS a S .

17) ( )X t and ( )Y t are zero mean and stochastically independent random process

having autocorrelation function ( )XX

R e

, ( ) cos 2YY

R 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 .

18) If ( )X t and ( )Y t are independent with zero means. Find the auto correlation

function of ( )Z t where ( ) ( ) ( )Z t a bX t cY t .

19) If ( ) 3cosX t t and ( ) 2cos2

Y t t

are two random processes

where is a random variable uniformly distributed in 0,2 . Prove that

0 0XX YY XY

R R R .

20) Two random process ( )X t and ( )Y t are given by ( ) cosX t A t ;

( ) sinY t A t where A and are constants and " " is a uniform random

variable over 0 to 2 . Find the cross – correlation function.

21) If ( )X t is a process with mean ( ) 3t and auto correlation

0.2, 9 4

XXR t t e

. Determine the mean, variance of the random variable

(5)Z X and (8)W X .


1) Prove that if the input ( )X t is WSS then the output ( )Y t is also WSS.

2) 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 0x

and 2( )

XXR e

. Find y

, ( )XX

S and

( )YY

S , if the system function is given by1

( )2

Hi

.



3) If ( )X t is a band limited process such that ( ) 0, XX

S , prove that

2 22 (0) ( ) (0)

XX XX XXR R R .

4) Let ( )X t be a random process which is given as input to a system with the system

transfer function 0 0( ) 1, H . If the autocorrelation function of the

input process is 0 . ( )2

N , find the auto correlation of the output process.

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

uniform distribution in , and ( )N t is a band limited Gaussian white noise

with a power spectral density 0( )2

NN

NS for

0 B and ( ) 0

NNS

,elsewhere. Find the power spectral density of ( )Y t , assuming that ( )N t and are

independent.

6) Consider a white Gaussian noise of zero mean and power spectral density 0

2

N

applied to a low pass RC filter whose transfer function is1

( )1 2

H fi fRC

. Find

the autocorrelation function of the output random process.

7) A WSS random process ( )X t with auto correlation ( )XX

R Ae

where A and

are real positive constants, is applied to the input of an linear time invariant (LTI)

system with impulse response ( ) ( )bt

h t e u t where b is a real positive constant.

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

8) An linear time invariant (LIT) system has an impulse response ( ) ( )t

h t e u t . Find

the output auto correlation function ( )YY

R corresponding to an input ( )X t .

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