6 Suppose that (x1,...,xn) is a sample from a Poisson () distribution with > 0 unknown. If we use the prior distribution for given by the Gamma (, ) distribution, then determine the posterior distribution of . 1. Suppose you toss a coin and put a Uniform[0.4, 0.6] prior on , the probability of getting a head on a single toss. (a) If you toss a coin n times and obtain n heads, then determine the posterior density of . (b) Suppose the true value of is, in fact, 0.99. Will the posterior distribution of ever put any probability mass around = 0.99 for any sample of n? (c) What do you conclude from part (b) about how you should choose a prior? 2. Under the set-up of Example 6 (in the lecture note of section 7.1.1) , i.e. Bernoulli random variables withBeta prior, derive the posterior mean of m where m > 0. 3. Under the set-up of Example 7 (in the lecture note of section 7.1.1), i.e. Normal random variables withNormal prior, determine the posterior distribution of the third quantile = + 0z0.25. Determine the posterior mode and posterior expectation of . 4. Suppose that (x1,...,xn) is a sample from the Exponential () distribution, where > 0 is unknown and Gamma(0, 0). Determine the mode of posterior distribution of . Also determine the posterior expectation and posterior variance of . 5. Consider the sampling model and prior as in Problem 3. (a) Suppose we want to estimate based upon having observed s = 1. Determine the posterior mode and posterior mean. Which would you prefer in this situation? Explain why. (b) Determine a 0.8 HPD region for based on having observed s = 1. 6. Consider the sampling model and prior as in Problem 5. Determine the mode of posterior distribution ofa future independent observation Xn+1. Also determine the posterior expectation of Xn+1 and posterior variance of Xn+1. 1