#$5($Sampling from Posterior Distribution$)$ Consider a Bayesian statistic model in which the data are i$.$i$.$d$.$ Poisson random variables with mean $.$ Suppose the prior of the parameter $>0$ is a $(,)$ with $,>0$ and its $PDF$ is

$f_0()=\frac{{}^{}}{()}*{}^{-1}{e}^{-},>0.$

$($a$)$ Derive the posterior distribution $f_1()$ given i$.$i$.$d$.$ data $x_1,$dots,$x_n.$

$($b$)$ Design an A$-$R algorithm to sample from the posterior distribution $f_1()$ and write down the pseudo codes.

$($c$)$ Implement the A$-$R algorithm. It should take the parameters $(,)$ of the prior distribution and the i$.$i$.$d$.$ data samples as input by the users. Illustrate the efficacy of your implementation using any examples you like and report the results.