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 $var[\frac{X_1+2X_2+3X_3}{X_1^2+X_2^2+X_3^2}]$