The lifetimes of certain components are supposed to follow a Weibull distribution with known shape parameter $\alpha=2$ . The probability density function of the lifetime distribution is  

$$
f(t)=\alpha\rho^{2}t\exp[-(\rho t)^{2}]
$$  

for $0<t<\infty$ .  

We will observe a sample of $n$ such lifetimes where $n$ is large.  

(a) Assuming that the prior density is nonzero and reasonably flat so that it may be disregarded, find an approximation to the posterior distribution of $\rho$ . Find an approximate 95% hpd interval for $\rho$ when $n=300$ ,$\sum\log(t)=1305.165$ and $\sum t^{2}=3161776$ .  
(b) Assuming that the prior distribution is a $\mathrm{gamma}(a,b)$ distribution, find an approximate $95\%$ hpd interval for $\rho$ , taking into account this prior, when $a=2$ ,$b=100$ ,$n=300$ ,$\sum\log(t)=1305.165$ and $\sum t^{2}=3161776$ . 