1. Let X1, . . . , Xn and Y1, . . . , Yn be samples from Poisson distributions with parameters 100(1 p) and 100p respectively. (a) Show that sumXi + sumYi is ancilliary for p. (b) In a particular experiment, the results are n = 5, sumXi = 46, sumYi = 72. Calculate the exact significance level of this result for the hypothesis p = 0.75. 2. Conditional probability: suppose that if = 1, then y has a normal distribution with mean 1 and standard deviation , and if = 2, then y has a normal distribution with mean 2 and standard deviation . Also, suppose P r( = 1) = 0.5 and P r( = 2) = 0.5. (a) For = 2, write the formula for the marginal probability density for y. (b) What is P r( = 1|y = 1), again supposing = 2? (c) Describe how the posterior density of changes in shape as is increased and as it is decreased. 3. Posterior inference: suppose you have a Beta(4,4) prior distribution on the probability that a coin will yield a head when spun in a specified manner. The coin is independently spun ten times, and heads appear fewer than 3 times. You are not told how many heads were seen, only that the number is less than 3. Calculate your exact posterior density (up to a proportionality constant) for . 4. Posterior distribution as a compromise between prior information and data: let y be the number of heads in n spins of a coin, whose probability of heads is . (a) If your prior distribution for is uniform on the range [0, 1], derive your prior predictive distribution for y, P r(y = k), for each k = 0, 1, , n. (b) Suppose you assign a Beta(, ) prior distribution for , and then you observe y heads out of n spins. Show algebraically that your posterior mean of always lies between your prior mean /+ ,and the observed relative frequency of heads, y/n . (c) Show that, if the prior distribution on is uniform, the posterior variance of is always less than the prior variance. 15. Normal distribution with unknown mean: a random sample of n students is drawn from a large population, and their weights are measured. The average weight of the n sampled students is y = 150 pounds. Assume the weights in the population are normally distributed with unknown mean and known standard deviation 20 pounds. Suppose your prior distribution for is normal with mean 180 and standard deviation 40. (a) Give your posterior distribution for . (Your answer will be a function of n.) (b) A new student is sampled at random from the same population and has a weight of y pounds. Give a posterior predictive distribution for y (Your answer will still be a function of n.) (c) For n = 10, give a 95% posterior interval for and a 95% posterior predictive interval for y