Applet Exercise In Exercise $16\mathrm{.}11,$ we found the posterior density for $,$ the mean of a Poisson$-$

distributed population. Assuming a sample of size $n$ and a conjugate gamma $(,)$ prior for

$,$ we showed that the posterior density of $|{\displaystyle \underset{?}{\overset{?}{}}}{y}_i$ is gamma with parameters ${}^{***}={\displaystyle \underset{?}{\overset{?}{}}}{y}_i+$

and ${}^{***}=\frac{}{n+1}.$ If a sample of size $n=25$ is such that ${\displaystyle \underset{?}{\overset{?}{}}}{y}_i=174$ and the prior

parameters were $(=2,=3),$ use the applet Gamma Probabilities and Quantiles to find a

$95\%$ credible interval for $.$

$16\mathrm{.}20$ Applet Exercise In Exercise $16\mathrm{.}12,$ we used a gamma $(,)$ prior for $v$ and a sample of size

$n$ from a normal population with known mean ${}_o$ and variance $\frac{1}{v}$ to derive the posterior for

$v.$ Specifically, if $u={\displaystyle \underset{?}{\overset{?}{}}}(y_i-{}_o{)}^2,$ we determined the posterior of $v|u$ to be gamma with

parameters ${}^{***}=(\frac{n}{2})+$ and ${}^{***}=2\frac{}{u+2}.$ If we choose the parameters of the prior

to be $(=5,=2)$ and a sample of size $n=8$ yields the value $u=\mathrm{.}8579,$ use the applet

Gamma Probabilities and Quantiles to determine $90\%$ credible intervals for $v$ and $\frac{1}{v},$ the

variance of the population from which the sample was obtained.

Solve only Q$16\mathrm{.}19$ and Q$16\mathrm{.}20$

$16\mathrm{.}11$ Let $Y_1,Y_2,$dots,$Y_n$ denote a random sample from a Poisson$-$distributed population with mean

$.$ In this case, $U={\displaystyle \underset{?}{\overset{?}{}}}{Y}_i$ is a sufficient statistic for $,$ and $U$ has a Poisson distribution with

mean $n.$ Use the conjugate gamma $(,)$ prior for $$ to do the following.

a Show that the joint likelihood of $U,$ is

$L(u,)=\frac{n^u}{u!{}^{}()}{}^{u+-1}$exp$[-\frac{}{\frac{}{n+1}}].$

b Show that the marginal mass function of $U$ is

$m(u)=\frac{n^u(u+)}{u!{}^{}()}(\frac{}{n+1}{)}^{u+}.$

c Show that the posterior density for $|u$ is a gamma density with parameters ${}^{**}=u+$

and ${}^{**}=\frac{}{n+1}.$

$16\mathrm{.}12$ Let $Y_1,Y_2,$dots,$Y_n$ denote a random sample from a normal population with known mean ${}_o$

and unknown variance $\frac{1}{v}.$ In this case, $U={\displaystyle \underset{?}{\overset{?}{}}}(Y_i-{}_o{)}^2$ is a sufficient statistic for $v,$ and

$W=vU$ has a ${}^2$ distribution with $n$ degrees of freedom. Use the conjugate gamma $(,)$

prior for $v$ to do the following.

a Show that the joint density of $U,v$ is

$f(u,v)=\frac{u^{(\frac{n}{2})-1}{v}^{(\frac{n}{2})+-1}}{()(\frac{n}{2}){}^{}{2}^{(\frac{n}{2})}}$exp$[-\frac{v}{\frac{2}{u+2}}].$

b Show that the marginal density of $U$ is

$m(u)=\frac{u^{(\frac{n}{2})-1}}{()(\frac{n}{2}){}^{}{2}^{(\frac{n}{2})}}(\frac{2}{u+2}{)}^{(\frac{n}{2})+}(\frac{n}{2}+).$

c Show that the posterior density for $v|u$ is a gamma density with parameters ${}^{**}=(\frac{n}{2})+$

and ${}^{***}=2\frac{}{u+2}.$