THEORY OF PROBABILITY DISTRIBUTIONS — DU B.Sc (hons) Statistics Previous Year Question Papers

DSC: Theory of probability Distributions (NEP) B.Sc. (Hons) Statistics (DSC) Attempt six questions in all, selecting tliree questions from each section. All questions/parts are of equal marks. All `notations used have their usual meanings. (a) Two independent random variables X and Y assume identical values -1,1, 3 with equal probabilities, then find Mx+v (t) and hence compute the distribution of x + Y. ®) Let X and Y.have joint probability density function Compute E(Y|X), and Var(E(Y|X)). (a) Find characteristic function of the random variable whose moments about origin are given by H! = (r+ I)! 9r, r = I,2 ,.... (b) Define cumulant generating function and show that all cumulants of Poisson distribution are equal. .(a) Let X and Y havejoint probability density function f (x, y) - Find the distribution of X -Y. (b) If x and Y are independent normal variates with means - I, 2 and variances 4 and 9respectively,thendetermineksuchthatp(3X+2Y<k)=P(X-4Y2k). 4. (a) For the probability mass function x= 0,1,2,... compute Fx (x) = P(X ≤x) and hence E(Fx (x)). 10) Show that - (1) ,. 5. (a) If X and Y are independent Poisson variates with means 2, and 2, respectively, then find conditional distribution of X|(X+Y = s). (b) Show that for Poisson distribution with mean (2), 14+1=2 It-and hence find first four central moments. 6. (a) In tri-variate distribution, derive an equation of plane of regression of Xi on X2 and (b) In tri-variate distribution show that 02 =ổ?® 7. (a) If a fair coin is tossed an even number (2k) times. Show that the probability of obtaining more tails than heads is (b) Show that for N(0,07) distribution Mar+1=0 and #2=1x3x5x...x (2x-1)03. 8. (a) Find mode of Binomial distribution with parameters (n, P). (b) What is mode of Binomial distribution in the following cases (i) Binomial distribution with parameters n = 12 and p = 1/3 and (ii) Binomial distribution with parameters n = 15 and p = 1/4.