Suppose that, given $\theta$, the random variable $X$ has $\mathbb{P}(X=k)=e^{-\theta} \theta^{k} / k !$, $k=0,1,2, \ldots .$ Suppose that the prior density of $\theta$ is $\pi(\theta)=\lambda e^{-\lambda \theta}, \theta>0$, for some known $\lambda(>0)$. Derive the posterior density $\pi(\theta \mid x)$ of $\theta$ based on the observation $X=x$.

For a given loss function $L(\theta, a)$, a statistician wants to calculate the value of $a$ that minimises the expected posterior loss

$$\int L(\theta, a) \pi(\theta \mid x) d \theta$$

Suppose that $x=0$. Find $a$ in terms of $\lambda$ in the following cases:

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

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