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PROBABILITY THEORY AND STOCHASTIC PROCESSES

Time: 3 hours Max Marks: 80

Answer any FIVE Questions

All Questions carry equal marks

1. (a) Define and explain following with example

i. Random Experiment

ii. Out comeiii. Trial and Event

(b) A die is tossed. Find the probabilities of the event A = {odd number showsup}, B = { number larger than 3 shous up }, A B and A B. [8+8]

2. (a) Define cumulative probability distribution function. Discuss distribution func-tion specific properties.

(b) The random variable X has the discrete variable in the set {-1,-0.5, 0.7, 1.5, 3}the corresponding probabilities are assumed to be {0.1, 0.2, 0.1, 0.4, 0.2}. Plotits distribution function and state is it a discrete or continuous distributionfunction. [8+8]

3. (a) State and prove properties of variance of a random variable.

(b) Let X be a random variable defined by the density functionfX(x) = (/16)cos(x/8), 4 x 4

= 0, elsewhere

Find E [3X] and E[X2]. [8+8]

4. (a) Define and explain joint distribution function and joint density function oftwo random variables X and Y.

(b) If the function f(x, y) = be2xcos(y/2), 0 x 1, 0 y

= 0, elsewhere

Where b is a positive constant is valid joint probability density function, findb. [8+8]

5. (a) Write the expression for expected value of a function of random variables andprove that the mean value of a weighted sum of random variables equals theweighted sum of mean values.

(b) X is a random variable with mean X = 3, variance 2X

= 2.

i. Determine the second moment of X about origin

ii. Determine the mean of random variable y = where y = -6X +22. [8+8]

6. Discuss in detail about:

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II B.Tech I Semester (R05) Supplementary Examinations, November/December 2011

Code No. : R5210402 R5

(Common to Electronics & Communication Engineering and Electronics & Computer Engineering)

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(a) First order stationary random process

(b) Second order & Wide - Sense Stationary Random Process. [8+8]

7. (a) A WSS random process X(t) has RXX () = A0

1 ||

t

= 0 else whereFind power density spectrum.

(b) RXX () =A20

2sin 0. Find Sxx () [8+8]

8. (a) Define the following random processes:

i. Band Pass

ii. Band limited

iii. Narrow band.

(b) A Random process X(t) is applied to a network with impulse response h(t)=u(t) exp (-bt) where b > 0 is constant. The Cross correlation of X(t) withthe output Y(t) is known to have the same form: RXY ( ) = u( ) exp(-bY):

i. Find the Auto correlation of Y(t)

ii. What is the average power in Y(t). [6+6+4]

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Code No. : R5210402R5

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