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Posted By: rama Member Level: Silver Posted Date: 11 Jun 2008
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2008 Anna University Electronics and Communication B.E/B.Tech DEGREE EXAMINATION, APRIL/MAY 2008 Question paper
B.E/B.Tech DEGREE EXAMINATION, APRIL/MAY 2008
Fourth Semester
Electronics and communication Engineering
MA1254-RANDOM PROCESSES
(Regulation 2004)
Time:Three hours Maximum:100 marks
Answer ALL questions
PART A –(10*2=20marks)
1. State the axioms of probability.
2. Define central moments of a random variable.
3. Write the moment generating function of Geometric distribution.
4. Write a note on functions of a random variable.
5. If the function f(x,y)=c(1-x)(1-y),0 function, find the value of c.
6. If Y=-2x+3, find the cov(X,Y).
7. What is a Markov process?
8. When is a stochastic process said to be ergodic?
9. What is meant by autocorrelation?
10. Write a note on a linear system.
PART – B(5*16=80 marks)
11. (a)(i) Members of a firm rent cars from three rental agencies. 60 percent
from agency 1, 30 percent from agency 2 and 10 percent from agency3.
If the percentages of the cars that tune-up are 9%,20% and 6%
respectively from agencies 1,2 and 3, what is the probability that it comes
from rental agency 2 if a rental car delivered to the firm needs tune-up?
(ii) Find the probability distribution of the total number of heads obtained in
four tosses of a balanced coin. Hence obtain the MGF of X, mean of X
and variance of X.
Or
(b)(i) In a small library there are 1000 books, among which 500 are scientific.
Among the scientific books are100 which belong to Engineering. Three
books are chosen at random, the chosen books being replaced each time.
What is the probability of getting (1)all three scientific books,(2) three
scientific books among which only one is an engineering book and (3) at
least one of three is an engineering book.
(ii) If X has the distribution function
0 x<1
1/3 1= x <4
½ 4 = x < 6
5/6 6= x < 10
1 x= 10
Find
(1) The probability distribution of X.
(2) P ( 2< X <6)
(3) Mean of X.
(4) Variance of X.
12. (a)(i) Obtain the MGF of Poisson distribution and hence compute the
first four moments.
(ii) State and explain the properties of Normal N( µ , s ² ) distribution.
Or
(b)(i) Find the variance of a random variable X which follows Negative
Binomial distribution.
(ii) State the density function of Weibull distribution and mention its
properties.
(iii) What is the variance of the random variable X which is Uniformly
distributed?
13. (a)(i) Given the joint probability density function of (X,Y):
f(x,y) = { e^-(x+y) x>0,y>0
0, elsewhere
Find the marginal densities of X and Y. Are X and Y independent?
(ii) Find the correlation between X and Y if the joint probability density
of X and Y is f(x,y) = { 2 for x>0, y>0 ,x+y<1
0 elsewhere
Or
(b)(i) Given the joint probability density
f (x,y) = { 2/3 (x +2y ) for 0< x<1 , 00 elsewhere
Find the marginal densities, conditional density of X given Y=y
and P ( X = ½ / y=1/2).
(ii) Let the joint probability distribution of X and Y be given by
x
-1 0 1
-1 1/6 1/3 1/6
y 0 0 0 0
1 1/6 0 1/6
Show that their covariance is zero even though the two random
variables are not independent.
14. (a)(i) Define a stationary stochastic process. If {X(t) ,t e T} is a process
with probability distribution.
P(X(t) = n) = { (at)^ n-1/ (1+ at)^n+1 , n=1,2…….
at/ 1+at n=0
Verify whether {X(t)} is a stationary process.
(ii) Write a detailed note on Normal process.
Or
(b)(i) Distinguish between ‘stationary’ and weakly stationary stochastic
processes. Give an example to each type. Show that Poisson
process is an evolutionary process.
(ii) Write a critical note on ‘sine wave’ process and its applications.
15.(a)(i) For a linear system with random input x(t), the impulse response h(t)
and output y(t), obtain the cross correlation function Ryx(T) and the
output autocorrelation function Ryy(T).
(ii) Starting your assumptions, discuss the relationship between cross
power spectrum and cross spectrum.
Or
(b)(i) Given a stationary random process X(t) = 10 cos (100t+?) where ? ? (-p, p)
followed Uniform distribution. Find the autocorrelation function of the
process.
(ii) Auto correlation function of an ergodic process {X(t) =X} is
Rxx(T)= { 1- |T| , |T|=1
0, otherwise . Obtain the spectral density of X.
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