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Posted By: b.thirumaya prabhu Member Level: Gold Posted Date: 21 May 2008
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2007 Anna University B.E B.E Electronics and Communication random process Question paper 07 Question paper
B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2003.
Fourth Semester
Electronics and Communication Engineering
MA 034 — RANDOM PROCESSES
Time : Three hours Maximum : 100 marks
Use of Statistical tables permitted.
Answer ALL questions.
PART A — (10 ? 2 = 20 marks)
1. A and B are events such that and . Find .
2. The p.d.f. of a random variable X is , find the p.d.f. of .
3. If X and Y are random variables having the joint density function , find .
4. Find the acute angle between the two lines of regression.
5. Define Markov chain and one–step transition probability.
6. State any two properties of a Poisson process.
7. Define autocorrelation function and prove that for a WSS process .
8. Define Cross–correlation function and state any two of its properties.
9. The power spectral density of a random process is given by Find its autocorrelation function.
10. Define a system. When is it called a linear system?
PART B — (5 ? 16 = 80 marks)
11. (i) Consider a random process where is wide sense stationary random process. is a random variable independent of and is distributed uniformly in and is a constant. Prove that is wide–sense stationary. (6)
(ii) Prove that the random process where A, w are constants and is a uniformly distributed random variable in is correlation ergodic. (10)
12. (a) (i) A, B and C, in order toss a coin. The first one to throw a head wins.
If A starts, find their respective chances of winning. (8)
(ii) A continuous random variable X has the p.d.f. . Find the rth moment of X about the origin. Hence find mean and variance of X. (8)
Or
(b) (i) Find the moment generating function of geometric distribution and hence find its mean and variance. (8)
(ii) A random variable X has the density function . Show that Chebychev’s inequality gives and show that the actual probability is . (8)
13. (a) (i) The joint density function of random variables X and Y is
, find the marginal and conditional
probability density functions. Are X and Y independent? (8)
(ii) 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 by more than 4? (8)
Or
(b) (i) Two independent random variables X and Y are defined by,
Show that and are uncorrelated. (8)
(ii) The joint p.d.f. of X and Y is given by , find the probability density function of . (8)
14. (a) (i) The autocorrelation function for a stationary process is given by . Find the mean value of the random variable and variance of . (6)
(ii) Prove that the random processes and defined by are jointly wide–sense stationary if A and B are uncorrelated zero mean random variables with the same variance. (10)
Or
(b) (i) If is a WSS process with autocorrelation function and if , show that
. (6)
(ii) Consider two random processes and where is a random variable uniformly distributed in . Prove that . (10)
15. (a) (i) Given that a process has the autocorrelation function
where and are real
constants, find the power spectrum of . (10)
(ii) A system has an impulse response , find the p.s.d. of the output corresponding to the input . (6)
Or
(b) (i) The cross–power spectrum of real random processes and is given by
Find the cross–correlation function. (8)
(ii) Show that where and are the power spectral density functions of the input and the output and is the system transfer function. (8)
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