Homework 12

Problem 1

Below is a sample from a Poisson distribution with rate \(\lambda\).

1. Find the maximum likelihood estimator of \(\lambda\).

2. Find the Bayesian estimator for \(\lambda\) if the prior distribution is exponential with rate 1 and using the posterior mean.

## x
##  5  6  7  8  9 10 11 12 13 14 15 16 19 
##  1  1  4  1  4  7  6  8  4  4  1  7  2

Problem 2

Let \((X_1, X_2, X_3)\) be a multinomial normal random vector with mean vector \((1, 1, 1)\) and variance-covariance matrix

\[ \begin{bmatrix} 1 & 0.5 & 0.1 \\ 0.5 & 2 & -0.3 \\ 0.1 & -0.3 & 3\\ \end{bmatrix} \]

Find an approximation to \(var[\frac{X_1+2X_2+3X_3}{X_1^2+X_2^2+X_3^2}]\)