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February 4th, 2013, 04:30 PM   #1
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A question about statistics...

Let X1, X2, , Xn be a random sample from a Poisson(?) distribution. Let be their sample mean and their sample variance.
a) Show that and both have a standard normal limiting distribution.
b) Find the limiting distribution of
c) Find the limiting distribution of


a) For , we know that = . Since
appraches N(0,1) in distribution by CLT, and since appraches 1 in probability, the whole thing approaches N(0,1).

For
, 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 and if there is a differentiable function g(x) at theta where the derivative at theta is not zero...then .

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
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