Statistical Methods — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

Find the marginal PDF of X and Y. Check whether the two random variables are independent. (7) 4. (a) A fair coin is tossed 10 times. Find: i. Probability of exactly 2 heads ii. Mean and variance 111. The mode(s) of the distribution and iv. Comment on the skewness of the distribution. (8) (b) Derive the recurrence relation for the probability mass function (PMF) of a binomial distribution. (4) (c) If X~Binomial (4,0.3), and Y~Binomial (6,0.3), find the PX+Y = 3). (3) 5. (a) Derive the cumulant generating function of a Poisson distribution. Using it, obtain Brand Bz. Hence, comment on the distribution's skewness. (8) (b) Show that the sum of independent Poisson random variables is Poisson. Explain why their difference does not follow a Poisson distribution. (7) 6. (a) Derive the moment generating function of a normal distribution. Using it, show that the odd-order standard moments are zero. (8) (b) If X is a normal variate with mean 10 and S.D. 3, find the probability that Also, find the value of 'a' such that P(X < a) = 0.25. (7) 7. (a) Define exponential distribution. Obtain the mean and variance of the distribution. If X has an exponential distribution, then for every constant a ≥ 0, show that P(Y ≤ x|X ≥ a) = P(X ≤ x), for all x, where Y = X- a. (8) (b) Define the Gamma distribution and derive its cumulant generating function. Also, explain its relationship with the exponential distribution. (7) 8. (a) "The mean of the binomial random variable is less than its variance", comment. (b) If X and Y are two independent normal random variables with parameters N(6, 4) and N(5,9), respectively. Find the distribution of 5X - ZY. (c) If X is Poisson (2) variate such that (iii) coefficient of skewness. (5)