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Re: Paper for Probability and random process
Here I am giving you question paper for Probability and random process examination of Anna university in a file attached with it 5. Define a wide sense stationary process. 6. Define a Markov chain and give an example. 8. Find the power spectral density function of the stationary process whose autocorrelation function is given by | |τ − e . 9. Define time-invariant system. 10. State autocorrelation function of the white noise. PART B — (5 × 16 = 80 Marks) 11. (a) (i) The probability mass function of random variable X is defined as 2 3 ) 0 ( C X P = = , 2 10 4 ) 1 ( C C X P − = = , 1 5 ) 2 ( − = = C X P , where 0 > C , and 0 ) ( = = r X P if 2 , 1 , 0 ≠ r . Find (1) The value of C (2) ) 0 / 2 0 ( > < < x X P (3) The distribution function of X (4) The largest value of X for which . (ii) If the probability that an applicant for a driver’s license will pass the road test on any given trial is 0.8. What is the probability that he will finally pass the test (1) On the fourth trial and (2) In less than 4 trials? (8) Or (b) (i) Find the MGF of the two parameter exponential distribution whose density function is given by ) ( ) ( a x e x f − − = λ λ , a x ≥ and hence find the mean and variance. (8) (ii) The marks obtained by a number of students in a certain subject are assumed to be normally distributed with mean 65 and standard deviation 5. If 3 students are selected at random from this group, what is the probability that two of them will have marks over 70? (8) 12. (a) (i) For the bivariate probability distribution of ) , ( Y X given below : Y X 1 2 3 4 5 6 0 0 0 1/32 2/32 2/32 3/32 1 1/16 1/16 1/8 1/8 1/8 1/8 2 1/32 1/32 1/64 1/64 0 2/64 Find the marginal distributions, conditional distribution of X given Y = 1 and conditional distribution of Y given X = 0. (8) (ii) Find the covariance of X and Y, if the random variable ) , ( Y X has the joint p.d.f. , ) , ( y x y x f + = , 1 0 ≤ ≤ x 1 0 ≤ ≤ y and 0 ) , ( = y x f , otherwise. (8) Anna university Probability and random process paper |
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