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 February 4th, 2013, 05:30 PM #1 Member   Joined: May 2012 Posts: 56 Thanks: 0 A question about statistics... Let X1, X2, …, Xn be a random sample from a Poisson(?) 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 $\sqrt{n}[\bar{X}^2-\lambda^2]$ 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. b) I'm a little confused about this one... 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 $\sqrt{n}(g(X_n)-g(\theta)) \rightarrow N(0,\sigma^2(g'(\theta))^2)$. 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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