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Homework Help: Proving a normal distribution

  1. Aug 22, 2010 #1
    Z=(-1/sqrt(n)) * sum from k=1 to n of [1+log(1-Fk)]

    Fk is a cumulative distribution function which is continious and strictly increasing.

    Show that as n->infinity, Z converges to a normal distribution with mean 0 and var 1

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    From Taylor series, log(1-x) = -sum from 1 to infinity of (x^n)/n but dont see how this can help at the moment
    Ive been looking for anything around the summation of c.d.fs but havent found anything so I think Im unaware of a few theorems which are essential to solving this. Any help appreciated. Been working on this for hours with no success.
     
  2. jcsd
  3. Aug 22, 2010 #2

    lanedance

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    can you exlpain what you mean by Fk? and how it depends on k?

    I understand its a cumulative distribution function, continuous & strictly increasing...

    however I may be missing something as I can't see where the randomness is coming into Z - otherwise I'd be thinking something long the lines of central limit theorem...
     
    Last edited: Aug 22, 2010
  4. Aug 22, 2010 #3
    Sorry, didnt make that clear. Xk is a sequence of independent random variables and Fk is its associated cumulative distribution function.

    So Fk is a sequence of cumulative distribution functions taking on values form 0 to 1. Can we simply look at it like this? And hence assume it is equivalent to a sequence of independent random variables with values 0 to 1?
     
  5. Aug 22, 2010 #4
    Some identities that are useful:

    Expected value of X = integration from 0 to inf of [1-F(t)] dt where F(t) is the cdf
    I need to obtain exp(-x^2/2) from the sum and log. Integrating x will obtain x^2/2 and taking the anti-log will obtain the exponential.

    Not sure how all this fits together though
     
  6. Aug 23, 2010 #5
    After looking at this more, it seems the moment generating function approach is the way to go.
    By obtaining the mgf and finding the mean and variance to be 0 and 1, we prove it is a normal distribution.
    Does anyone know if this is acceptable?
     
  7. Aug 23, 2010 #6

    lanedance

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    i still don't understand your explanation of Fk

    is there only a single cdf F(x)?

    so for every Xk has the same cdf Fi(x) = P(Xi<=x)?

    if so, then the map between Xk & Fk is one to one monotonic and the random varable Fk, becomes a uniform random variable between 0 and 1
     
  8. Aug 23, 2010 #7
    No, there is a cdf for each x.

    Fk(xk) for k=1 to n
    F1(x1), F2(x2) etc

    Sorry, it's the lack of latex that makes it hard to show subscript
     
  9. Aug 23, 2010 #8
    I looked at taking the mgf thinking if I can show the mean to be 0 and variance 1 from this approach and hence prove normality

    g(t) = 1(1/sqrt(n) sum(from 1 to inf) exp(tk) [1+log(1-Fk(Xk))]

    but this approach goes nowhere
     
  10. Aug 23, 2010 #9

    lanedance

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    even if they're different you should be able to show any Fk(x=Xk) represents a uniform random variable on [0,1], by definition of the cdf - have you tried using that property?
     
  11. Aug 23, 2010 #10
    As n tends to infinity, the cumulative distribution function Fk(Xk) tends to F(X)

    I think this is ok. I don't see how to show this tends to a normal distribution though.

    I think I'm making this much harder than it is. Don't see anything close in any textbook or on the web!
     
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