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dharavsolanki

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**Central Limit Theorem Variation for Chi Square distribution?**

If this question fits into Homework Help, please move it over there. I'm not too sure.

I encountered the following problem:

An experiment E is performed n times. Each repetition of E results in one and only one of the events Ai, i = 1, 2, 3, ..., k. Suppose that p(Ai) = pi and Let ni be the number of times Ai occurs among the n repetitions of E, n1 + n2 + ... +nk = n.

Also, let D^{2}= Summation of i from 1 to k of [tex]\frac{(n_i - np_io)^2}{np_io}[/tex]

If n is sufficiently large and if p_{i}= p_{i}o then show that the distribution of D^{2}has approximately the chi square distribution with k-1 degrees of freedom.

Now, this problem seems fairly similar to a simple proof the central limit theorem. I am damn sure that this problem involves finding the mgf of D2, evaluating it and saying that it is the same as the mgf of a chi square function.

Can you help me out with setting up the equation? that'll be a big help! Thank you!

I've given it an attempt, but one attempt at setting up the mgf is all that i ask. Also, if there is any other way which does not involve mgf (i being wrong), please mentione that! Thank you!

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