Suppose that X is a discrete random variable with PX1 theta

Suppose that X is a discrete random variable with P(X=1) = theta and P(X=2)=1-theta. Three independent observations of X are made: x1=1, x2=2, x3=2

a) Find the method of moments estimate of Theta
b) What is the likelihood function?

c)What is the maximum likelihood estimate of theta?

d) if capital theta has a prior distribution that is uniform on [0,1] what is its posterior density?

Solution

a)

E(X) = ?(1) + (1 ? ?)(2)

= 2 ? ?.

or equivalently,

? = 2 ? E(X).

The method of moments estimate of ? isgiven by

?\' = 2 ?