Problem 1

Say \(X_1,..,X_n\) is a sample from a mixture of normal distributions with density

\[f(x;\mu,\sigma)=\frac1{2\sqrt{2\pi}}\exp\{-\frac12 x^2\}+\frac1{2\sqrt{2\pi\sigma^2}}\exp\{-\frac1{2\sigma^2} (x-\mu)^2\}\]

Let \(L(\mu,\sigma)\) be the likelihood function. Show that \(\max \{L(\mu,\sigma):-\infty<\mu<\infty;0<\sigma<\infty\}=\infty\).

(This is an example where the likelihood function is quite badly behaved)

Problem 2

Say \(X\sim Beta(a, 1)\). Find the posterior distribution when the prior is \(\pi(a)=1/a,a>1\).

Problem 3

Below is data from a geometric random variable with rate p, that is \(P(X=k)=p(1-p)^{k-1}, p=1,2,...\). Find the Bayesian point estimator of \(p\) using the mean of the posterior distribution and the prior \(p\sim U[0,1]\).

x Counts
1 64
2 20
3 9
4 5
5 2