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Math Help - Proving a distribution convergs to a standard normal distribution

  1. #1
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    Proving a distribution convergs to a standard normal distribution

    Y has a binomial distribution based on n trials and success probability p. Then the point estimator (pn=Y/n) is an unbiased estimator of p. Prove that the distribution (pn-p)/sqr((pn*qn)/n) converges to a standard normal distribution.

    I am supposed to the following theorem to prove it:
    Suppose that Un has a distribution function that converges to a standard normal dist function as n-->∞. If Wn converges in probability to 1, then the dist. function Un/Wn converges to a Standard normal dist. function.


    I am unsure of how to approach this so any help is greatly appreciated.
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  2. #2
    Super Member Anonymous1's Avatar
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    Use MGF. It suffices to show that Your MGF \rightarrow MGF_{Standard-Normal} as n\rightarrow \infty

    I actually recently did a similar problem, so if you show me what you have, I can work through it with you.

    Start by deriving the MGF of of the distribution in question.

    [EDIT]Oops, I was thinking you were trying to prove something similar to the CLT.
    Last edited by Anonymous1; March 29th 2010 at 02:28 PM.
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  3. #3
    MHF Contributor matheagle's Avatar
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    It seems that you just want to apply slutsky's theorem.

    The CLT gives you {P_n-p\over {\sqrt{pq\over n}}}\to N(0,1)

    All you need is the WLLN to finish...

    {P_n\over p}\buildrel P \over \to 1 and {1-P_n\over 1-p}\buildrel P \over \to 1

    which gives you

    \sqrt{{P_n(1-P_n)\over p(1-p)}}\buildrel P \over \to 1
    Last edited by matheagle; March 29th 2010 at 02:36 PM.
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