Let y₁,..., yn be a sample from a Poisson distribution with mean λ, where A is given a Gamma(a, ß) prior distribution. (a) It is observed that y₁ = y2 =.= yn = 0, and we take a = 1, B = 1. i. What is the posterior distribution for X? ii. What is the posterior mean? iii. What is the posterior median and an equal tail 95% credible interval for X (without using R)? (b) Show that if a new data-point x is generated from the same Poisson distribution, the posterior predictive probability that x = 0 is p(x = 0|y) = n+1 n+2 (c) Now suppose that we have general y₁,..., yn, a and 3; and that again x is a new data-point from the same Poisson distribution. i. Find the mean and variance of x. ii. Derive the full posterior predictive distribution for x.

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