It is required to estimate the unknown parameter $\theta$ after observing $X$, a single random variable with probability density function $f(x \mid \theta)$; the parameter $\theta$ has the prior distribution with density $\pi(\theta)$ and the loss function is $L(\theta, a)$. Show that the optimal Bayesian point estimate minimizes the posterior expected loss.

Suppose now that $f(x \mid \theta)=\theta e^{-\theta x}, x>0$ and $\pi(\theta)=\mu e^{-\mu \theta}, \theta>0$, where $\mu>0$ is known. Determine the posterior distribution of $\theta$ given $X$.

Determine the optimal Bayesian point estimate of $\theta$ in the cases when

(i) $L(\theta, a)=(\theta-a)^{2}$, and

(ii) $L(\theta, a)=|(\theta-a) / \theta|$.