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

Advanced Probability Distributions, DSC - 8 B.Sc. (Hons.) Statistics (Under NEP - UGCF) 2. Attempt six questions in all selecting two questions from each section. Section: A l. (a) Derive the cumulant generating function of negative binomial distribution and hence obtain first four cumulants, p, md Br. (b) Let X is a negative binomial variate with p.m.f., Show that the moment recurrence formula is: In an infinite sequence of independent Bemoulli trails with constant probability ,p, of success for each trail, find the mean and variance of the number of trails required to get the first success. Explain how and under what conditions the hyp€rgeometric distribution can be approximated by the binomial distribution. (9,7) Define multinomial distribution. obtain the formula for the correlation coefticient between two random variables following a multinomial distribution. Two discrete random variables X and Y have the joint probability distribution: For x, y : 0, 1,2, ...,9 and 0 < x+y < 9. Show that (i) Show that the marginal distribution of X is Binomial with parameters n = 9 (ii) Show that the conditional distriburion ofy given X:3, is also binomial with parameters n= 6andp = 7/2. (7,g) 4 (a) If X has a uniform distribution in f0,ll. Find the pdfof-2logX. and also idenrily the distribution. L (b) Suppose that probability function of a random variable X is given by: 1(2) =10; f*e-;forx> 0 and k>0 (i) Find the moment generating function of this distribution. (ii) Determine the mean and variance of this distribution using moment generating function. 5. (a) Suppose that X is a random variable for which E(X) = u and Var(X) = o2. Further suppose that Y is uniformly distributed over the interval (a, B). Determine a and B, so that E(X) = E(Y) and Var(X) = Var (Y). (b) Show that the mean value of positive square root of a r (4) variate is y (4+=) /Г(u). Hence prove that mean deviation of a N (4, o2) variate from its mean is V2/n. 6. (a) If X & Y are independent Gamma variates with parameter f & v respectively. Find the distribution of U =X+Y, and Z = X*y Also show that they are independently distributed. (b) Define beta distribution of first kind. Compute rt"moment and harmonic mean for beta distribution of first kind. (8,7) 7. (a) Find the characteristic function of standard Laplace distribution and hence find its mean and standard deviation (b) Let X has a standard Cauchy distribution. Find the p.d.f for X2 and identify the distribution. (7,8) 8. (a) A sample of n values is drawn from a population with p.d.f. If X is the mean of the sample, show that neX is a y (n) variate and prove that E(X) = - and S.E. (X) = Uri (b) Obtain the moment generating function of Logistic distribution and hence find its mean and variance. (8,7) 9. (a) If X1, X2, X3,.... Xn be ini.d. random sample from a population with continuous density.