1. One of my prized possessions is my 'simplified quantum coin', described by P(X = 1|p) = p, P(X =0|p) = 1 - p, where X is the number of heads in a toss of the coin, and p is a random variable with P(p = 1/4) = 1/4 and P(p = 3/4) = 3/4.
(a) What is the probability of a 'head' in a single toss of this coin?
(b) If the coin is tossed and a 'head' results, what is the probability that p is 3/4?
2. Consider a random sample X1, , Xn from a Uniform distribution on (0, θ). Derive a likelihood ratio test of H0 : θ = θo versus HA : θ ≠ θ0. Show that the rejection region is of the form X(n) ≤ cθo or X(n) > θo and find c for level α = 0.05.