2. Let $X_1, \dots, X_n$ be a random sample from the distribution with the p.d.f.
$f(x|\theta) = (1/2)e^{-(x-\theta)/2} I(0 < x < \infty)$, $-\infty < \theta < \infty$. (20)
a) Find the likelihood function $L(\theta)$ and the MLE $\hat{\theta}_n$ of $\theta$.(10) (hint: draw the graph of $L(\theta)$)
b) Find the constant $a_n$ so that $\hat{\theta}_n - a_n$ is an unbiased estimator of $\theta$ for $n \ge 1$. (10)
(hint: $X_i = U_i + \theta$, $i = 1, \dots, n$; $U_1, \dots, U_n$ is a random sample from the distribution with the p.d.f.
$f(u) = (1/2)e^{-u/2}$, $0 < u < \infty$, $U_{(1)} = min(U_1, \dots, U_n)$, $p(U_{(1)} < u) = 1 - e^{-nu/2}$, $0 < u < \infty$)
