Categories: Second Year First Semester (2-1)

Code No: R1621045

II B. Tech I Semester Supplementary Examinations, May – 2018

RANDOM VARIABLES & STOCHASTIC PROCESSES (Electronics and Communication Engineering)Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A

3. Answer any FOUR Questions from Part-B

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART ?A

1. a) In a single throw of three dice, find the probability of getting the same number

on three dice. (2M)

b)Define moment generating function of a random variable X. (2M)

c) X and Y are two independent random variables with E[X] = 4, E[Y] = 6. Find

E[4X-2Y] (2M)

d)When a random process is called SSS process? Explain (3M)

e) Determine whether the power density spectrum shown below is valid or not?

2

6 2

3 3

?

? ?

+ +(3M)

f)Define effective noise temperature. (2M)

PART -B

2. a) Define a discrete random variable and discuss the characteristics of Poisson

random variable using its probability density and distribution functions. (5M)

b)Define probability distribution function and write its properties. (4M)

c) A random variable X has pdf shown below. i) Find the value of k. ii) Find P(1/4

{

0 1

0( )

kx x

X elsewhere

f x

=

(5M)

3. a) A random variable X has a probability density

{( /16)cos( /8) 4 4

0( )

x x

X elsewhere

f x

p p – = =

=

Find its variance. (7M)

b) A random variable X has pdf f

X(x) = (1/b)e

-(x-a)/b

. Find its characteristic

function . (7M)

4. a) Find the marginal densities of the joint density

2( ) 2 2 3 3( , )

0

XY

b x y x and y

x y

elsewhere

f

? + –
=

?

?(7M)

b) Two random variables X and Y have joint characteristic function

?

XY(?

1,

?

2

) = exp(-2?

1

2

-8?

2

2

). Show that X and Y are zero mean uncorrelated

random variables. (7M)

1 of 2

SET – 1

R16

Code No: R1621045

II B. Tech I Semester Supplementary Examinations, May – 2018

RANDOM VARIABLES & STOCHASTIC PROCESSES (Electronics and Communication Engineering)Time: 3 hours Max. Marks: 70

Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A

3. Answer any FOUR Questions from Part-B

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

PART ?A

1. a) In a single throw of three dice, find the probability of getting the same number

on three dice. (2M)

b)Define moment generating function of a random variable X. (2M)

c) X and Y are two independent random variables with E[X] = 4, E[Y] = 6. Find

E[4X-2Y] (2M)

d)When a random process is called SSS process? Explain (3M)

e) Determine whether the power density spectrum shown below is valid or not?

2

6 2

3 3

?

? ?

+ +(3M)

f)Define effective noise temperature. (2M)

PART -B

2. a) Define a discrete random variable and discuss the characteristics of Poisson

random variable using its probability density and distribution functions. (5M)

b)Define probability distribution function and write its properties. (4M)

c) A random variable X has pdf shown below. i) Find the value of k. ii) Find P(1/4

{

0 1

0( )

kx x

X elsewhere

f x

=

(5M)

3. a) A random variable X has a probability density

{( /16)cos( /8) 4 4

0( )

x x

X elsewhere

f x

p p – = =

=

Find its variance. (7M)

b) A random variable X has pdf f

X(x) = (1/b)e

-(x-a)/b

. Find its characteristic

function . (7M)

4. a) Find the marginal densities of the joint density

2( ) 2 2 3 3( , )

0

XY

b x y x and y

x y

elsewhere

f

? + –
=

?

?(7M)

b) Two random variables X and Y have joint characteristic function

?

XY(?

1,

?

2

) = exp(-2?

1

2

-8?

2

2

). Show that X and Y are zero mean uncorrelated

random variables. (7M)

1 of 2

SET – 1

R16

Code No: R1621045

5. a)Write short notes on Gaussian random process. (7M)

b) Given the random process X(t) = Acos?

o

t + Bsin?

o

t, where ?

o

is a constant

and A, B are uncorrelated zero ,mean random variables with equal variances.

Prove that X(t) is wide sense stationary. (7M)

6. Derive the relationship between cross correlation function and cross power

spectrum. (14M)

7. a)Write in detail about resistive noise source. (8M)

b) Obtain the mean value of the response of a LTI system excited by random

process X(t). (6M)

2 of 2

SET – 1

R16

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