1. Feb 4, 2013

### Artusartos

1. The problem statement, all variables and given/known data

Let X1, X2, …, Xn be a random sample from a Poisson(λ) distribution. Let $\bar{X}$ be their sample mean and $S^2$ their sample variance.
a) Show that $\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}$ and $\frac{\sqrt{n}[\bar{X}-\lambda]}{S}$ both have a standard normal limiting distribution.
b) Find the limiting distribution of $\sqrt{n}[\bar{X}-\lambda]^2$
c) Find the limiting distribution of $\sqrt{n}[\bar{X}^2-\lambda^2]$

2. Relevant equations

3. The attempt at a solution

a) For $\frac{\sqrt{n}[\bar{X}-\lambda]}{S}$, we know that $\frac{\sqrt{n}[\bar{X}-\lambda]}{S}$ = $(\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma})(\frac{\sigma}{S})$. Since
$\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma}$ appraches N(0,1) in distribution by CLT, and since $(\frac{\sigma}{S})$ appraches 1 in probability, the whole thing approaches N(0,1).

For
$\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}$, we get the same result...since the mean is equal to the variance in a poisson distribution.

c) From a theorem in my textbook, I know that if $\sqrt{n}(X_n - \theta) \rightarrow N(0,\sigma^2)$ and if there is a differentiable function g(x) at theta where the derivative at theta is not zero...then $\sqrt{n}(g(X_n)-g(\theta)) \rightarrow N(0,\sigma^2(g'(\theta))^2)$.