Let $X_{1}, \ldots, X_{n}$ be independent samples from the Poisson distribution with mean $\theta$.

(a) Compute the maximum likelihood estimator of $\theta$. Is this estimator biased?

(b) Under the assumption that $n$ is very large, use the central limit theorem to find an approximate $95 \%$ confidence interval for $\theta$. [You may use the notation $z_{\alpha}$ for the number such that $\mathbb{P}\left(Z \geqslant z_{\alpha}\right)=\alpha$ for a standard normal $\left.Z \sim N(0,1) .\right]$

(c) Now suppose the parameter $\theta$ has the $\Gamma(k, \lambda)$ prior distribution. What is the posterior distribution? What is the Bayes point estimator for $\theta$ for the quadratic loss function $L(\theta, a)=(\theta-a)^{2} ?$ Let $X_{n+1}$ be another independent sample from the same distribution. Given $X_{1}, \ldots, X_{n}$, what is the posterior probability that $X_{n+1}=0$ ?

[Hint: The density of the $\Gamma(k, \lambda)$ distribution is $f(x ; k, \lambda)=\lambda^{k} x^{k-1} e^{-\lambda x} / \Gamma(k)$, for $x>0$.]