The survival times, in months, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential $(\lambda)$ distribution. We observe the survival times of $n$ such patients. Our prior distribution for $\lambda$ is a $\mathrm{gamma}(a,b)$ distribution.  

(a) Prior beliefs are expressed in terms of the median lifetime, ${\boldsymbol{r}}n$ . Find an expression for ${\boldsymbol{r}}n$ in terms of $\lambda$ .  
(b) In the prior distribution, the lower 5% point for ${\boldsymbol{r}}n$ is 6.0 and the upper 5% point is 46.2. Find the corresponding lower and upper $5\%$ points for $\lambda$ . Let these be $k_{1},\ k_{2}$ respectively.   
(c) Let $k_{2}/k_{1}=r$ . Find, to the nearest integer, the value of $\nu$ such that, in a $\chi_{\nu}^{2}$ distribution, the $95\%$ point divided by the 5% point is $r$ and hence deduce the value of $a$ .  
(d) Using your value of $a$ and one of the percentage points for $\lambda$ , find the value of $b$ .  
(e) We observe $n\,=\,25$ patients and the sum of the lifetimes is 502. Find the posterior distribution of $\lambda$ .  
(f) Using the relationship of the gamma distribution to the $\chi^{2}$ distribution, or otherwise, find a symmetric $95\%$ posterior interval for $\lambda$ .  
Note: The R command qchisq(0.025,nu) will give the lower 2.5% point of a $\chi^{2}$ distri  
bution on nu degrees of freedom. 