We are interested in the parameter, $\theta$ , of a binomial $(n,\theta)$ distribution. We have a prior distribution for $\theta$ with density  

$$
f^{(0)}(\theta)=\left\{\begin{array}{c c}{k_{0}\{\theta^{2}(1-\theta)+\theta(1-\theta)^{2}\}}&{(0<\theta<1)}\\ {0}&{(\mathrm{otherwise})}\end{array}\right..
$$  

(a) i. Find the value of $k_{0}$ .  

ii. Find the prior mean of $\theta$ .iii. Find the prior standard deviation of $\theta$ .(b) We observe $x$ , an observation from the binomia $(n,\theta)$ distribution.  

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