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
1. What is the maximum likelihood estimate of theta?

2. If upper case theta has a prior distribution that is uniform on [0,1], what is its posterior density?

Solution

To find estimate of theta we need to equate theoretical
moments with empirical moments.

In this concrete case we equate the expected value with
the sample mean.
........_
EX = X

EX = 1*? + 2*(1 - ?) = ? + 2 - 2? = 2 - ?
_
x = (1 + 2 + 2)/3 = 5/3

Then

2 - ? = 5/3 or

estimate of theta ?^ = 1/3.

Likelihood function :

L(x1, x2, ..., xn, ?) = Pr[X = x1]*Pr[X=x2]*... *Pr[X=xn]

L(1, 2, 2, ?) = Pr[X = 1]*Pr[X=2]*Pr[X=2] =

= ?*(1- ?)*(1 - ?) = ?*(1 - ?)