Results 1 to 6 of 6

Thread: law of large number

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    12

    law of large number

    I have problem with the following question

    Let $\displaystyle \xi_1, \xi_2,...$ be independent and identically distributed random variables with $\displaystyle E|\xi_i|< \infty $. Show that

    $\displaystyle E\left(\xi_1|S_n,S_{n+1},...\right)=\frac{S_n}{n} \ \ \ (a.s)$
    where $\displaystyle S_n= \xi_1+...+\xi_n$

    I proceed this way:
    I want to know wether this is true or not.

    $\displaystyle S_n = E(S_n|S_n)=E\left(S_n|S_{n},S_{n+1},S_{n+2},...\ri ght)$ ???

    Now since $\displaystyle \xi_i$ are independent and identically distributed, then
    $\displaystyle E(\xi_1)=E(\xi_2)=...=E(\xi_n) $

    $\displaystyle E \left(S_n|S_{n},S_{n+1},S_{n+2},...\right)=n.E\lef t(\xi_1|S_{n},S_{n+1},S_{n+2},...\right)$

    so finally
    $\displaystyle E\left(\xi_1|S_n,S_{n+1},...\right)=\frac{S_n}{n} \ \ \ (a.s)$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I'll use X's instead

    $\displaystyle E(S_n|S_n)=S_n$

    So $\displaystyle E(X_1+\cdots +X_n|S_n)=S_n$

    Thus $\displaystyle E(X_1|S_n)+\cdots +E(X_n|S_n)=S_n$

    But each of these are the same, producing $\displaystyle nE(X_1|S_n)=S_n$
    Last edited by matheagle; Jan 20th 2011 at 10:09 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    12
    Thanks for your reply,

    I want to know what about this, is this true ?

    $\displaystyle E(S_n|S_n)=E\left(S_n|S_{n},S_{n+1},S_{n+2},...\ri ght)$

    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Both are S_n, as long as you condition on S_n, you know S_n, the rest of that sequence is unnecessary.

    E(X|X)=X=E(X|X,Y)
    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
    Same problem : http://www.mathhelpforum.com/math-he...-v-136142.html

    And I don't agree with you going to this step :

    $\displaystyle E \left(S_n|S_{n},S_{n+1},S_{n+2},...\right)=n.E\lef t(\xi_1|S_{n},S_{n+1},S_{n+2},...\right)$

    equality of the expectations doesn't mean equality of the conditional expectations.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    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
    As for proving that $\displaystyle E[S_n|S_n,S_{n+1},\dots]=E[S_n|S_n]$, remember that if we have two sigma-algebras (or sigma-algebras generated by random variables) $\displaystyle \mathcal G, \mathcal H$ such that $\displaystyle \mathcal H\subset \mathcal G$, then for a rv X, $\displaystyle E[E[X|\mathcal G]|\mathcal H]=E[X|\mathcal H]$

    now consider $\displaystyle \mathcal H=\sigma(S_n)$ and $\displaystyle \mathcal G=\sigma(S_n,S_{n+1},\dots)$ and you have your equality.

    (why is there the inclusion ? because if $\displaystyle S_n$ is a measurable function of $\displaystyle \xi_1,\dots,\xi_n$ and $\displaystyle (S_n,S_{n+1},\dots)$ is a measurable function of $\displaystyle \xi_1,\dots,\xi_n,\dots$)

    If you don't know what a sigma-algebra is, then just forget my post
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prime factoring a large number.
    Posted in the Algebra Forum
    Replies: 5
    Last Post: Sep 10th 2011, 04:10 AM
  2. Using sqrt(d) factor large number
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: Jan 9th 2011, 04:43 AM
  3. [SOLVED] Strong law of large number
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Oct 31st 2010, 03:16 AM
  4. Large Number
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: Jul 10th 2009, 08:02 PM
  5. Large number probality question
    Posted in the Statistics Forum
    Replies: 1
    Last Post: Jun 12th 2008, 02:06 PM

Search Tags


/mathhelpforum @mathhelpforum