5. Waiting for a Random Coin to Land Heads Let X be a random proportion. Given X = p, let T be the number of tosses till a p-coin lands heads. a) Let P(X = 1/10) = 1/4, P(X = 1/7) = 1/4, and P(X = 1/3) = 1/2. Find E(T). b) Find E(T) if X has the beta (r, s) density for some r > 1. Simplify all integrals and Gamma functions in your answer. c) Let X have the beta (r, s) density. For fixed k > 0, find the posterior density of X given T = k. Identify it as one of the famous ones and state its name and parameters.