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1. Let Y1,Y2, Yn be independent and identically distributed as N(0,0²) for some o 0. a) Find the MLE for o². Justify. b) Why is an MLE (in general) usually considered to be a good estimator for a parameter? c) Find the UMVUE for o². Justify. 2. The Beta-Binomial model. Let X1 X, Xn be independent Bernoulli random variables with probability of success (i.e. of being 1) given by p. a) Assume the prior distribution of p is given by a beta(a: ß) distribution. Derive the posterior distribution of p. b) Romney (1999) looks at the level of consensus among 24 Guatemalan women on whether they think polio is non-contagious. The survey data are given below: 1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 where 1 indicates that the respondent believes polio to be non-contagious and 0 indicates that the respondent believes polio to be contagious. Let p denote the probability of each woman believing polio to be non-contagious. Apply the beta-binomial model with a beta(1; 1) prior to find the posterior distribution of p. c) For the above data, find a Bayes point estimate and a 90% credible interval for p.3. Suppose X - exponential(1.1) and Y - exponential(i.2) where X and Y are independent and 21, 1.2 >0. Define Z = min(X.Y) and let W = 1 ifX < Y, and W = 0 otherwise. a) Derive the probability mass function of W. b) What is the distribution of Z? justify your answer. c) Consider two simple random samples (X1.X2. Xn) from exponential(21) and (Y1,Y2 Y.) from exponential(i.2), and the two samples are independent of each other. n Define zi = min(Xi,Yi) for i = 1,2, n. What is the distribution of = Also, describe its limiting distribution under appropriate standardization4. A retailer buys items from a supplier; each item can be either acceptable or defective, and separate items are independent. a) Suppose the probability of each item being defective is 0.1. What is the probability that there are 6 defective items in a lot of 25? b) Suppose there are 6 defective items in a lot of 25. If7 items are randomly sampled without replacement from the lot, what is the probability of finding no defective item among the 7 items? c) The lot will be unacceptable if more than 5 items are defective. Suppose the retailer selects randomly K items and decides to accept the lot if there is no defectivei item in the sample. How large does K have to be to ensure that the probability that the retailer accepts an unacceptable lot is less than 0.10? 5. Let Y1.Y2, Yn be independent and identically distributed as Poisson(i)) for some N>. a) Show that the most powerful level-a test of Ho: N=1 vs. Ha: N=2 rejects Ho when >Y; >c for some c. b) Argue that your test from part (a) is UMP for testing Ho: N=1 vs. Ha: 1. c) One can test Ho: N=RO vs. Ha: NHO for any choice of no: the GLR test rejects Ho if >Y; S C1 or C2 for C1, c2 depending on re. a, and n. Assuming that you have a computer program that can find C1 and C2 for every choice of No, a., and n. How might you construct a conservative 95% confidenceregion for A given a value for £Yi ? 6. Suppose X1 - gamma( 0.1, 1) and X2 - gamma(a), 1), where X1 and X2 are independent and 0.1, 0.2 > 0. Justify your answers to each of the following. a) Define V = X1 + X2. What is the distribution of V? b) Define U = X1/(X1 + X2). What is the distribution of U? c) What is the conditional distribution of U given V?

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