Can someone answer the question in chap 8 for the book Mathe

Can someone answer the question in chap 8 for the book Mathematical Statistics and Data Analysis by Rice? the question is 5.

I need help with b and c. the question is Suppose that X is a discrete random variable with P(X=1) = theta and P(X=2) = 1-theta. 3 indepedent observations of X are made x1 = 1, x2=2, x3=2

b) what is he likelihood function?

c) what is the maximum likelihood estimate of theta

thanks

Solution

a) 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.

b) 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 - ?)