Consider a sample from a distribution with PDF
$$f(x) = \begin{cases} \frac{1}{2}(1+\theta x), & -1 \leq x \leq 1\\ 0, & otherwise \end{cases} $$
find the maximum likelihood estimator of $\theta $.
I know that
$$ L(p) = f(x_{1},\theta) . f(x_{2},\theta) ... f(x_{n},\theta)$$ Therefore, $$ L(p) = \frac{1}{2}(1+\theta x_{1}). \frac{1}{2}(1+\theta x_{2})...\frac{1}{2}(1+\theta x_{n}) = \frac{1}{2}^n.(1+\theta x_{1}).(1+\theta x_{2})...(1+\theta x_{n}) $$
Now how can I formalize the remaining product part so that I can obtain MLE of $\theta$ ?
Thank you for your help.