Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Central Limit Theorem Variation for Chi Square distribution?

  1. Apr 25, 2010 #1
    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:

    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!
    Last edited: Apr 25, 2010
  2. jcsd
  3. Apr 25, 2010 #2
    Re: Central Limit Theorem Variation for Chi Square distribution???

    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 Yij, where Yij = 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!!!
  4. Apr 25, 2010 #3


    User Avatar
    Homework Helper

    Re: Central Limit Theorem Variation for Chi Square distribution???

    Begin by looking at the distribution of each

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

    and not that even though there are [tex] n [/tex] of them they satisfy one linear relationship.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook