STOCHASTIC PROCESSES — DU B.Sc (hons) Statistics Previous Year Question Papers

2372013503 (NEP-UGCF) Stochastic Processes 2. There are TWO sections in this paper. 1a)Letx„,fo"even,fakesvalues+land-1,eachwithprobabilityi,andformdd,take values Va,a , with probabilities i + respectively (a is real number >-I and l+fl,1+c' i 0,I ). Further let Jr„ 's be independent. Is {Xn, n 2 1} covariance stationary? b) In a series of Bemoulli trials with probability of success p,let X denote the number of failures preceding the first success and Y, the number of failures following the first success and . Hence preceeding the second success, then show that the p.g.f. of Z=X+Y is P(s,s)= (T=)2 obtain an expression for P(Z = r) . c) Suppose a ball is thrown at random to orie of the r cells. Let Jrn (nil) be said to be in state k(=1,2„ . .,r) if after n throws k cells are occupied. Obtain the transition probability matrix of d) In the classical ruin problem, prove that a fair game remains fair and no unfair game can be changed to unfair game. e) Prove that the interval between two successive occurrences of a Poisson process {N(t), t≥ 0} having parameter 2 has a negative exponential distribution with mean f f) Let X be a non-negative integral valued random variable with probabilityp(X = n) = Pre n = 0,1,2.., and probability generating function P(s) = Zk=oPksk. Find the generating function for P(X > n). g) Construct the transition probability matrix when game starts with $6 and gambler with initial capital $3 starts the game and his probabilities of winning, losing and drawing a game are 2a) Let an denote the number of ways in which score n can be obtained by throwing a die any number of times. Show that the gif, of fan) is given by {1-5-3-5'-s"-5-s"}'-1. b) Let S= X, +X,+...+Xy, where X,'s are ii.d. random variables and N is a random variable independent of X,'s. Show that Also show that for a compound geometric distribution where k is the compounding parameter. (8,7) 3a) Let X be a zero- one truncated Poisson variate having the p.m.f.: Obtain the probability generating function of r.v. X and hence obtain mean and variance of X. Further, if X, and X, are iid. having p.m.f. same as that of r.v. X, then find the probability generating function of X, + X,. b) Let SIV = jr, +X2 +...LrIV , where IV has Poisson distribution with mean cz . If Jr,. 's have i.i.d. binomial distributions with parameters in and p, then show that i) Thejoint distribution of s„ and Iv has the probability mass function ii) Cov(IV,SIT) =map 4a) On lst January 2024 each of the three leading automobile companies Mahindra, Hyundai, Tata Motors launched a new model of electric car. When the new models were launched each company had an equal share of the market but during the year the following changes in the market share were observed. Mahindra retained 90% of its customers but lost 3% to Hyundai and 7% to Tata Motors. Hyundai retained 700/o of its customers and lost 10% to