Exercise 1: Let a population Y with distribution given by
$f_Y(y) = (1 - \theta) + 2\theta y$, $0 < y < 1$,
where $\theta$ is a parameter in the interval $(-1, 1)$.
We select from this distribution a sample of size $n$: $Y_1, \dots, Y_n$.
(5) Find the estimator $\hat{\theta}$ the MLE of $\theta$
(6) Find $I_n(\theta)$ the fisher information of $\theta$
(7) Compute the CRLB of $\theta$
(8) Is $\hat{\theta}$ a MVUE of $\theta$? why?
(9) Find the asymptotic distribution of $\hat{\theta}$
(10) For $n$ large, deduce a 95% confidence interval of $\theta$
