Q.2 Answer the following questions. (14 marks) a. Given {Nn)n21 number of successes in a Bernoulli stochastic process. Compute and P{N1 = a, N3 = b, N4 = c}, where choose a, b, c E Zt. b. Compute Var(N2 + dN3) where choose d E Z+. c. Discuss the difference between the Bernoulli stochastic process and random walk process. d. Given X1, X2, . .., Xn be iid random variables. If E (X;) = 0, Vi = 1, n and E(X?) = ?. Let Sn = So + Et=1X; , where So is a constant and Vn = Et_1 07 be the variance of S, then show that S? - Vn is a martingale w.r.t Sn. e. What is the difference between the Random walk process and the Brownian motion process? Give your answer in a maximum of 10 lines. f. Let {Beltzo is a standard Brownian motion process. Find distribution of By + Bt , where s t. Is geometric Brownian motion a martingale process? Justify your answer. g. What was the aim and scope of this course