The Poisson distribution has the probability density function (pdf)
$$
f(x \mid \lambda)=\frac{\lambda^{x} e^{-\lambda}}{x !}, \text { for } x=0,1, \ldots
$$
(a) Show that the Poisson distribution is an exponential family, identifying the parameters $\theta$ and $\phi$ as well as the functions $b(\theta)$ and $a(\phi)$.
(b) Obtain the canonical link. Show your work.
(c) Obtain the variance function. Show your work.

Let $X_{1}, \cdots, X_{n}$ be independent random variables from a Poisson distribution with the pdf given in Question $1 .$
(a) What is the log-likelihood for this example?
(b) What is the Fisher information for this example?
(c) Find the MLE of $\lambda$ and its asymptotic distribution.

Consider a random sample $X_{1}, \cdots, X_{n}$ satisfying $X_{i} \stackrel{d}{=} \operatorname{pois}(\theta)$, i.e., $f\left(x_{i} \mid \theta\right)=\frac{\theta^{x_{i}} e^{-\theta}}{x_{i} !}$. To assess an estimator $\hat{\theta}=t\left(X_{1}, \cdots, X_{n}\right)$ of $\theta$ we use the loss function $L(\hat{\theta} ; \theta)=\frac{(\hat{\theta}-\theta)^{2}}{\theta} .$ We assume a gamma prior pdf $\theta \stackrel{d}{=} \operatorname{gamma}(\beta, \kappa)$ with known $\beta$ and $\kappa$, i.e. $p(\theta)=\frac{1}{\beta^{\kappa} \Gamma(\kappa)} \theta^{\kappa-1} e^{-\theta / \beta} ; \theta>0$
(a) Show that the Bayes estimator under the given loss function is $\hat{\theta}=\left(E\left[\theta^{-1} \mid \mathrm{x}\right]\right)^{-1}$.
(b) What is the posterior distribution of $\theta$. Show your work.
(c) Find a closed form for $\hat{\theta}$ in (a) by using the result of (b). Show your work. [Hint: $\Gamma(n)=(n-1) !]$

You only have access to uniform random number generator. For a random $X, f(x) \propto \log (1+x)$ if $x \in[0,2]$, and 0 else. Provide a rejection sampling algorithm (or pseudocode) to sample $X$. Your algorithm should be the most efficient algorithm (i.e., one that minimises the probability of rejection).

Consider a Metropolis-Hastings (MH) algorithm where you propose $\theta_{n}$ from a $\operatorname{Normal}\left(\theta_{o}, 1\right)$ distribution (where $\theta_{o}$ is the current sample) and accept with probability $\min \left(1, \frac{\exp \left(\frac{\theta_{o}^{2}}{2}\right)}{\exp \left(\frac{\theta_{n}^{2}}{2}\right)}\right)$. What is the stationary distribution? Briefly describe the MH algorithm to simulate samples from that stationary distribution.

You want to sample from $f(\theta, \mu \mid x) \propto \exp \left(-\frac{(\theta-\mu-1)^{2}}{(\mu+1) x^{2}}\right)$, where $\theta$ is real-valued and $\mu \in\{0,1\}$ and $\mathrm{x}$ is known.
(a) Give the conditional distributions of $\theta \mid \mu, x$ and $\mu \mid \theta, x$, including their parameters.
(b) Briefly describe a Gibbs sampling algorithm for sampling $(\theta, \mu)$, and how you would use it to estimate the mean of $\frac{\theta}{\mu}$.