We are interested in the parameter $\lambda$ of a Poisson(λ) distribution. We have a prior distribution for $\lambda$ with density  

$$
f^{(0)}(\lambda)=\left\{\begin{array}{l l}{{0}}&{{(\lambda<0)}}\\ {{k_{0}\lambda^{3}e^{-\lambda}}}&{{(\lambda\geq0)}}\end{array}\right..
$$  

(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 $x_{1},\ldots,x_{n}$ which are independent observations from the Poisson(λ) distribution.  

i. Find the likelihood function.   
ii. Find the posterior density of $\lambda$ .  
iii. Find the posterior mean of $\lambda$ .  