ELEMENTS OF STOCHASTIC PROCESS — DU B.Sc (hons) Statistics Previous Year Question Papers

3. Attempt four questions from the remaining questions. (a) If G(s) is the p.g.f. of X, what are the expectation and variance of X? Show the derivation. (b) Let (Xn, n ≥ 1) be a sequence of uncorrelated random variables with mean 0 and variance 1. Is {Xn, n ≥ 1) a covariance stationary process? Explain. (d) Let n=01 n-1+90s,(n=1,2,...), where do, agare constants and Xn, n= 0,1,2,.. are ii.d. random variables with mean 0 and variance o?. Is {Yn,n ≥ 1) covariance stationary? Explain. 2. (a) Let SN = X1 + X2 + ... +Xu where Xi's are i.i.d. random variables with p.g.f. P(s) and N is a random variable independent of Xi's with p.g.f. G(s). What is the p.g.f. of SN. Give a detailed proof. (b) Using the result in part (a) obtain the p.g.f. of a Compound Poisson distribution. Clearly indicate the concerned notations. 3. (a) What is a Markov chain? Explain with examples. What do you mean by the transition probability matrix of a Markov chain? What are its properties? (b) Consider a Markov chain (X, n ≥ 0} such that Xn assumes values -1 and 1 with conditional probabilities: and initial distribution P{X, = 1} = P1 = 1 - P(X. = - 1}. i) E{Xn} ii) Var{ Xn} ill) first-order Cov Xn-1, Xn} iV) Cov{Xa-1, Xn} when a=b. 4. (a) Obtain the p.g.f. of the following distributions. Hence obtain their expectation and variance. (i) Binomial (n,p) distribution. (ii) Geometric (p) distribution. (b) Explain clearly what you mean by a reducible Markov chain. 5. (a) Define a Poisson process. State the postulates that characterize a Poisson process and derive the probability law of a Poisson process (N(t), t ≥ 0]. Also show that Also, prove that the mean and variance of N(t) are both equal to it. b) Suppose the number of customers arriving at a shop follows a Poisson process with rat per hour. Each customer independently buys with probability O.t Fin the dailyit end yes of the number of buyers in 2 hours 6. (a) If {N(t)} is a Poisson process, then show that the autocorrelation coefficient between (b)Let [Xn, n ≥ 0} be a Markov chain having state space S={0, 1, 2, 3}and transition matrix as: i) Is the chain irreducible? ii) Find the nature of all states.