Results 1 to 9 of 9

Math Help - Central limit proof

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    11

    Central limit proof

    Z=(-1/sqrt(n)) * sum from k=1 to n of [1+log(1-Fk)] where 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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2010
    Posts
    11
    Just to clarify, Fk is the cumulative distribution function. So for k=1 to n, F1, F2, F3 etc are separate cumulative distribution functions
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Since Fk is a cdf, where is the variable ? You gotta have something like Fk(x) So could you clarify this ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2010
    Posts
    11
    Yes sorry. It's Fk(Xk).

    So this will take value from 0 to 1 and is uniform
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hm I don't understand. You mean it's the cdf taken at the value of a random variable, Xk ?
    Isn't it that Fk is the cdf of Xk ?

    And what kind of random variable is Xk ?

    Why do you say Fk is uniform ? Would it mean that it's the cdf of a uniform distribution ? Over what interval ?


    Is there the exact writing of the problem somewhere ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2010
    Posts
    11
    Xk is an independent random variable and Fk is it's associated cumulative distribution function.

    Sorry if I didn't make this clear. Missed this from the original question description.

    So, Fk(Xk) will become a uniform distribution from 0 to 1 since it has equL likelihood of any vale from 0 to 1 given that Xk is independent and random. At least this is what I'm thinking now
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Yes sorry that's right, given that Fk is strictly increasing (quite important assumption), U_k=F_k(X_k) follows a uniform distribution over (0,1)

    And we know (we can check it with cdf) that -\log(1-U_k) follows an exponential distribution with parameter 1. And its expectation is 1.
    So the expectation of -(1+\log(1-U_k)) is 0.

    And here, we have -\frac{1}{\sqrt{n}}\sum_{k=1}^n 1+\log(1-U_k)=\frac{1}{\sqrt{n}}\sum_{k=1}^n -(1+\log(1-U_k))... so, can you apply the CLT here ?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Aug 2010
    Posts
    11
    This is good. Thanks.
    I took the approach of calculating the moments of a log uniform by taking the standard approach but replacing the function, x, with ln(x) ie

    1/(b-a) integral from a to b of ln(x)

    Using this gave me -1 for the expected value and 1 for the variance.

    I didnt think of using the quartile function and this looks to be a better approach. How do you calculate the variance though? Understand the mean is 1/parameter = 1
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Find the variance of the exponential distribution of parameter 1 (that is 1 ;D) and since the random variables are independent, the variance of the sum is the sum of the variances !

    The cdf is sometimes much more easier to use than calculating all the moments ^^ (use the inversion method)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Central Limit Theorem to proof that...
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: December 20th 2011, 01:45 PM
  2. Central limit theorem
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: January 4th 2010, 06:15 AM
  3. Please help: central limit example
    Posted in the Advanced Statistics Forum
    Replies: 9
    Last Post: December 17th 2009, 08:54 AM
  4. Central Limit Theorem.
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: May 27th 2009, 05:59 AM
  5. Using the Central Limit Theorem to prove a limit
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 23rd 2009, 12:09 PM

Search Tags


/mathhelpforum @mathhelpforum