Introduction

Verify the expressions for $\pi (\lambda \mid \bar{x})$ and $\delta^{k}(\bar{x})$ in Example 1.3.

Give examples of pairs of values $(a, b)$ for which the beta density $B(a, b)$ is

In Example 1.5, if $p$ has the improper prior density $\frac{1}{p(1-p)}$, show that the posterior density of $p$ given $x$ is proper, provided $0<x<n$.

In Example 1.5, find the Jeffreys prior for $p$ and the associated Bayes estimator $\delta_{\Lambda }$.

For the estimator $\delta_{\Lambda }$ of Problem 1.4,

In Example 1.5, find the Bayes estimator $\delta$ of $p(1-p)$ when $p$ has the prior $B(a, b)$.

For the situation of Example 1.5, the UMVU estimator of $p(1-p)$ is $\delta^{\prime }=[x(x- 1)] /[n(n-1)]$ (see Example 2.3.1 and Problem 2.3.1).

In analogy with Problem 1.2, determine the possible shapes of the gamma density $\Gamma (g, 1 / \alpha ), \alpha , g>0$.

Let $X_{1}, \ldots , X_{n}$ be iid according to the Poisson distribution $P(\lambda )$ and let $\lambda$ have a gamma distribution $\Gamma (g, \alpha )$.

For the situation of the preceding problem, solve the two parts corresponding to Problem 1.5(a) and (b).

In Problem 1.9, if $\lambda$ has the improper prior density $d \lambda / \lambda$ (corresponding to $\alpha =g=$ 0 ), under what circumstances is the posterior distribution proper?

Solve the problems analogous to Problems 1.9 and 1.10 when the observations consist of a single random variable $X$ having a negative binomial distribution $N b(p, m)$, $p$ has the beta prior $B(a, b)$, and the estimand is