These are past year paper questions. Answers were not provided, so i would know how to do them.

Question 1. First Step Analysis (20 marks) You toss a fair coin repeatedly. (10 marks) (a) What is the expected number of tosses until the pattern HT appears for the first time? (10 marks) (b) What is the expected number of tosses until n consecutive heads appears for the first time?Question 2. Markov Chain (20 marks) 8 is the state space of a Markov chain. (8 marks) (a) Let S = {0,1} be the statespace of a time-homogeneous Markov chain. Suppose that IP' (Xn = OIXn_1 -- 1) _. ,8 E (0, l). Derive the probability distribution for the number of consecutive time steps spent in state 1, including the rst step. What is the name of this distribution? (12 marks) (b) Consider a game server that can go off-line with probability p E (0, 1) and can remain online with probability 1 p on any given day. Assume that the random time N it takes to x the server has distribution P(N=k)=(1)k'1, k>1 With ,6 6 (O, 1). We let Xn = O to denote that the server is online on day n and Xn = 1 to denote that it is off-line. Compute the probability that the server is online in the long run, in terms of the parameters 5 and p. Question 3. Branching Process (20 marks) The distribution of offspring for each individual is lP'({=0)=O.2, ]P'(=1)=0.5, IP(=2)=0.3. Let Xn denote the total number of individuals at generation 71, with X0 = 1. (6 marks) (a) Compute the mean and variance of X2. (4 marks) (b) Compute the expected number of offsprings at generation 10. (10 marks) (0) Calculate the probability of extinction. Question 4. Birth and Death Process (20 marks) (20 marks) Let (Xthemn be a birth and death process on {0,1,2} with birth and death parameters A0 = 2oz, /\\1 = a, A2 = 0 and M0 = 0, M1 = , #2 = Z. Determine the stationary distribution of (Xt)tR+. Question 5. Expectation (20 marks) Let N be the number of distinct words that Victor Hugo knew, and assume these words are numbered 1 to N. Suppose for simplicity that Victor Hugo wrote only 2 novels, A and B. The novels are reasonably long and they are the same length. Let X j be the number of times that word j appears in novel A, and Y) be the number of times it appears in novel B, for 1 S j S N. Assume that X; and Y3- follows a Poisson distribution with the same parameters. (6 marks) (a) Let the number of occurrences of the word \"pieuvre\" in the two novels be independent Poisson distributed with parameter A. Show that the probability that \"pieuvre\" is used in novel B but not in novel A is A2 A3 A4 ,\\ 6 (A+ET1T+W) Hint: We have (6 marks) (b) Assume that A from part (a) is unknown and is itself taken to be a random variable with probability density function f). Let X be the number of times the word \"pieuvre\" appears in novel A and Y be the corresponding value for novel B. Assume that the conditional distribution of X , Y given A is that they are independent Poisson distributed with parameter A. Show that the probability that \"pieuvre\" is used in novel B but not in novel A is the alternating series 1P(X=l)IP(X=2)+IP'(X=3)]P(X=4)+... Hint: Condition on A and use part (a). (8 marks) (c) Assume that every word's numbers of occurrences in A and B are distributed as in (b), where A may be different for different words but f; is xed. Let VVJ- be the number of words that appear exactly 3' times in novel A. Show that the expected number of distinct words appearing in novel B but not in novel A is IE[W1] E[W2]+E[W31E[W4]+m