We are interested in the mean, $\lambda$ , of a Poisson distribution. We have a prior distribution for $\lambda$ with density  

$$
f^{(0)}(\lambda)=\left\{\begin{array}{c c}{{0}}&{{(\lambda\leq0)}}\\ {{k_{0}(1+\lambda)e^{-\lambda}}}&{{(\lambda>0)}}\end{array}\right..
$$  

(a) i. Find the value of $k_{0}$ .ii. Find the prior mean of $\lambda$ .iii. Find the prior standard deviation of $\lambda$ .  
(b) We observe data $x_{1},\ldots,x_{n}$ where, given $\lambda$ , these are independent observations from the Poisson $(\lambda)$ distribution. i. Find the likelihood. ii. Find the posterior density of $\lambda$ .iii. Find the posterior mean of $\lambda$ . 