Central Limit Theorem Variation for Chi Square distribution?

[dharavsolanki]

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 D² = Summation of i from 1 to k of $$\frac{(n_i - np_io)^2}{np_io}$$

If n is sufficiently large and if p_i = p_io then show that the distribution of D² 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!

 

[dharavsolanki]

I have seen a proof of this theorem, which has been proved by assuming the value of the parameter k = 2. Essentially, it just means that the theorem has been proved using ony the asumption tht only two events may occur.

p1 + p2 = 1 and n1 + n2 = n

Using this and substituting in the value of D2, we arrive at a uncture where n1 is defined as sum of j from 1 to n of Y_ij, where Y_ij = 1 is A₁ occurs on the jth repetition and 0 elsewhere.

Now, central limit theorem is used over the variable n1, and if n is large, it has approximately a normal distribution.


I distinctly remember someone mentioning the use of Principle of Mathematical Induction to solve this problem. Can anyone of you solve this problem using the PMI? Will be a big help! No mgf involved till now!

 

[statdad]

Begin by looking at the distribution of each

$$\frac{(n_i - np_io)}{\sqrt{np_io}}$$

and not that even though there are $$n$$ of them they satisfy one linear relationship.