Question 1.  [8 MarkS]  Suppose X ~ Poisson(λ), where λ > 0 is the mean parameter of X , and Y is a Bernoulli random variable with P[Y = 1] = p and P[Y = 0] = 1 _ p.

(a) Calculate the moment generating function of Y .

(b) Assuming X and Y are independent, find the moment generating function of Z  = X + Y .  By differentiating the moment generating function of Z an appropriate number of times , find the mean and variance of Z.

(c) Determine the probability mass function of the conditional distribution Y IZ = z .

(d) Determine the probability mass function of the conditional distribution XIZ = z .

Question 2.   [10 MarkS]  A manuscript is sent to a typing unit to be typed by one of three typists, Typist 1, Typist 2 or Typist 3. The probability distribution of the number of errors for Typist j is Poisson with mean parameter λj  for j = 1, 2, 3. Assume that each of the three typists is equally likely to be asked to do this typing job and let N denote the number of typing errors that are present in the completed job.

(a) Determine the probability mass function of N. Calculate (i) E[N] and (ii) Var(N).

(b) Suppose that there are n typing errors, i.e. N = n. Calculate the probabilities P[Typist j did the typingIN = n],     j = 1, 2, 3.

(c) Suppose that λ 1  < λ2 < λ3 . If N = 0, which typist is most likely to have done the typing? Give justification.

(d) Still assuming λ 1   < λ2 < λ3 , which typist is most likely to have done the typing if N is large? What is the probability that the most likely typist in fact did the typing and what is the limiting value of this probability as n → o? Briefly explain your reasoning.

Question 3 [12 Marks]

(a) A security officer with n different keys wants to open a door. One of the keys is known to be the correct key. Let N denote the number of trials needed to open the door and assume that candidate keys are selected at random. Calculate E[N] and Var(N) in the following two cases: (i) unsuccessful keys are eliminated from further consideration; and (ii) unsuccessful keys are NOT eliminated from further selection.

(b) An urn contains n balls labelled 1 to n. Balls are drawn one at a time and then put back in the urn. Let M denote the number of draws before some ball is chosen more than once. Find the probability mass function of M.

Hint for part (b): First find the distribution of M for a few small values of n and then try to identify the pattern for general n.

Question 4.  [10 MarkS]  Two people, Frank and Maria, play the following game in which they throw two dice in turn. Frank’s objective is to score a total of 5 while Maria’s objective

is to throw a total of 8. Frank throws the two dice first. If he scores a total of 5 he wins the game but if he fails to score a total of 5 then Maria throws the two dice. If Maria scores 8 she wins the game but if she fails to score 8 then Frank throws the two dice again. The

game continues until either Frank scores a total of 5 or Maria scores a total of 8 for the first time. Let N denote the number of throws of the two dice before the game ends.

(a) What is the probability that Frank wins the game?

(b) Given that Frank wins the game, calculate the expected number of throws of the two dice, i.e. calculate E[NIF] where F is the event {Frank wins}.

(c) Given that Frank wins the game, calculate the conditional variance Var(NIF).

(d) Calculate the unconditional mean E[N].      (e) Calculate the uncoditional variance Var(N).