How Do Sample Means and Variances of Poisson Distributions Approach Normality?

[Artusartos]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

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

b) I'm a little confused about this one...

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

So I just need to use this theorem, right? And in this case g(x)=x^2.

Do you think my answer for a), and c) are correct? Also, can you give me a hint for b)?

Thanks in advance

 

[mighty2000]

!
Thank you for your interesting question. I have studied and worked with various statistical distributions, including the Poisson distribution. Based on my knowledge and experience, I would like to offer some suggestions to help you with your problem.

For part a), your approach seems to be correct. By using the Central Limit Theorem, we can show that both \frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}} and \frac{\sqrt{n}[\bar{X}-\lambda]}{S} have a standard normal limiting distribution. As for part b), we can use the same theorem to find the limiting distribution of \sqrt{n}[\bar{X}-\lambda]^2. Remember that the variance of a Poisson distribution is equal to its mean, so you can use this to simplify your solution.

For part c), you are correct in using the theorem you mentioned. You just need to find the derivative of g(x)=x^2 at \theta=\lambda, which is 2\lambda. Then you can plug this into the theorem to find the limiting distribution of \sqrt{n}[\bar{X}^2-\lambda^2].

I hope this helps and good luck with your problem! If you need any further clarification, please don't hesitate to ask. Keep up the good work in your studies.