Probability Theory & Stochastic Processes B.Tech Question Paper : vardhaman.org

College : Vardhaman College Of Engineering
Degree : B.Tech
Department : Electronics & Communication Engineering
Semester : III
Subject : Probability Theory & Stochastic Processes
Document type : Question Paper
Website : vardhaman.org

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Vardhaman Probability Theory Question Paper

B. Tech III Semester Regular Examinations, December – 2013
(Electronics and Communication Engineering)
Date : 13 December, 2013
Time : 3 Hours
Max. Marks : 75
Answer ONE question from each Unit
All Questions Carry Equal Marks
All parts of the question must be answered in one place only

Related : Vardhaman College Of Engineering Strength Of Materials-I B.Tech Question Paper : www.pdfquestion.in/6325.html

Unit – 1

1. a) There are three unbiased coins and one biased coin with head on both sides. A coin is chosen at random and tossed 4 times. If head occurs all the four times, what is the probability that the biased coin has been chosen? 8M
b) Find the characteristic function of an exponential distribution x e x f x otherwise Hence find the mean and variance 7M
2. a) A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known that B is more likely to fail (probability 0.06) if A failed.

i. What is the probability of an accidental missile launch?
ii. What is the probability that A will fail if B has failed?
iii. Are events “A fails” and “B fails” statistically independent? 8M
b) A random variable X has a probability density Find 7M
i. Mean value
ii. its second moment

Unit – 2

3. a) List the properties of the joint distribution function for two random variables X and Y? 6M
b) Two random variables X and Y have the density function 6M
i. Find all the first and second order moments 7M
ii. Find the covariance
iii. Are X and Y uncorrelated?

4. a) Given the joint function 8M
i. Find a constant so that this is a valid joint density function.
ii. Find { 2 2 P 0.5b x y 0.8b }.

b) Two random variables X and Y have means E[X] = 1 and E[Y] = 2 and, variancesVar[X] = 4 and Var[Y] =1 and a correlation coefficient is 0.4 .New random variables W and V are defined by V=-X + 2Y W = X+3Y Find i) mean ii) variance iii) correlation coefficient VW ? of V and W. 7M

Unit – 3

5. a) Explain cross correlation function and prove its properties. 9M
b) Consider the random process, X (t) given by ? ? cos?2 ? c X t ? A ? f t ?? , where A and c f are constants and ? is a random variable that is uniformly distributed over the interval (-p, p ). Determine the auto correlation function of X(t). 6M
6. a) Let two random process X(t) and Y(t) be defined by Where A and B are random variables assumed to be zero mean and un correlated. Also A and B are assumed to have variance ?2 . Find the cross correlation function Rxy

b) A random process is defined as X (t) = A cos(pt). Where A is a Gaussian random variable with zero mean and variance 2 A ? .
i. Find the density functions X(0 )and X(1)
ii. Is X(t) stationary in any sense? 8M

Unit – 4

7. a) Given that the process X(t) has an auto correlation 0 ( ) cos( ) XX R Ae where A, a > 0 and 0 ? are real constants, find power spectrum of X(t). 8M
b) Consider a bridge circuit with white noise of spectral density N0 as input. Find the spectral density of the output for the following fig.1

8. a) Derive the relation between cross power density spectrum and cross correlation function of a random process. 8M
b) Define cross power spectral density of two continuous time random process X(t) and
Y(t). Prove that
i. ( ) ( ) XY YX S w ? S ?w
ii. * ( ) ( )
XY XY S ?w ? S w
iii. If X(t) and Y(t) are orthogonal then (w ) = 0

Unit – 5

9. a) Obtain an expression to find Noise figure of a multistage amplifier with a neat figure. 8M
b) Develop a mathematical model of narrow band Noise and discuss its related properties. 7M
10. a) Define in-phase component and quadrature-phase component. Show that a narrowband noise process can be expressed as in-phase and quadrature components of it.8M
b) Define average noise figure and average noise temperature. Derive mathematical expressions for both. 7M