Results 1 to 7 of 7

Math Help - Central Limit Theorem

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    218

    Central Limit Theorem

    From my understanding, according to the central limit, T = X_1 + X_2 + ... + X_n should behave (roughly) like an N(0,1) distribution for a large enough n.

    I'm trying to show this by simulation. I created 1000 X_i iid ~U[0,1]. So according to CLT, T~N(0,1). But how would I show this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    According to Central Limit Theorem, \frac{T-mean(T)}{sd(T)} follows a N(0,1) distribution for large n. Also, if X_i's are iid U(0,1) random variables, then their sum do not follow N(0,1) distribution.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    the average of uniforms, when standardized should be approximately normal.
    same goes for the sum
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Oct 2009
    Posts
    218
    so \frac{T - \frac{n}{2}}{n*\sqrt{\frac{1}{12}}} is approximetaly N(0,1)?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    since these are iid with a second moment we have

    {\bar X-{1\over 2}\over \sqrt{1\over 12n}}\to N(0,1)

    we have {n\bar X-{n\over 2}\over n\sqrt{1\over 12n}}\to N(0,1)
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Oct 2009
    Posts
    218
    So if I were to plot that, I would get a rouch bell curve, correct? But I try to do this on R

    for (i in 1:1000){
    + T=(sum(runif(1000))-(1000/2))/1000*sqrt(1/12*1000)
    + V[i]=T
    + }

    I create runif(1000) creates 1000 U[0,1] and T is me trying to standardize it using CLT. I do this 1000 times, but I dont get a bell curve when I plot V.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Did you try....... {T_n-{n\over 2}\over \sqrt{n\over 12}}

    because yours is wrong, that n in the denominator is incorrect.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: July 21st 2010, 04:27 PM
  2. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 4th 2008, 12:36 AM
  3. central limit theorem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 31st 2008, 04:31 PM
  4. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 18th 2008, 09:49 PM
  5. Central Limit Theorem
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 2nd 2008, 12:34 PM

Search Tags


/mathhelpforum @mathhelpforum