Suppose that X is a discrete random variable with P(X = 1) = θ and P(X = 2)
= 1−θ. Three independent observations of X are made: x1 = 1, x2 = 2, x3 = 2.
a. Find the method of moments estimate of θ.
b. What is the likelihood function?
c. What is the maximum likelihood estimate of θ?
d. If has a prior distribution that is uniform on [0, 1], what is its posterior
density?