Let $X_1, ...., X_n$ be iid random variables from the following distribution
| $x$ | $-1$ | $0$ | $1$ |
|---|---|---|---|
| $P(X=x)$ | $\theta / 3$ | $(1- \theta)$ | $2\theta/3$ |
for $0\leq \theta \leq 1$.
Based on a sample of size $n$, find
(a) suficient and complete statistics for $\theta$.
(b) MLE for $\theta$
My try:
We got that
$\prod^n_{i}f(x_i;\theta)=\prod^n_{x_i=-1}\theta/3\prod^n_{x_i=-1}(1-\theta)\prod^n_{x_i=-1}2\theta/3=L(\theta)$
for b) I know that I have to find the máximum of the function $L(\theta)$ but I'm not sure how to do this with these discrete random variables. Any suggestions would be great!