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July 17th, 2012, 12:01 AM  #1 
Newbie Joined: Jul 2012 Posts: 3 Thanks: 0  Law of Large Numbers and the Expected Value
Hi, I am trying to understand the following issue: suppose we have n random variables Xi IID, with E(Xi) = u. We know that the Law of Large Numbers says that Xn > u, n > Inf, where Xn is the mean of n random variables. We also know that if we have a random value Xn = (X1+X2+X3...Xn)/n and take the expected value of it, i.e., E(Xn), we get u. So, the way I see it, both results suggest the same outcome, convergence towards u. Now, does one prove the other? Does one suggest the other? Or do they just mean different things? I mean, if E(Xn) = u can be calculated by using the Expected Value "arithmetics" (linearity, etc), then it would suggest that we can expect Xn to produce u (or close to it). But when does LLN come into play at this point? 
July 17th, 2012, 12:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,276 Thanks: 516  Re: Law of Large Numbers and the Expected Value
The expected value is a statement about ensemble average (m) in probability theory, while the law of large numbers refers to the average (A) of n trials. A is a random variable itself with E(A) = m, but A can have any value depending on the underlying distribution. The law of large numbers then has A > m as the number of trials becomes infinite. Note: there are two laws of large numbers, weak and strong, with different definitions for A > m. 
July 18th, 2012, 05:49 AM  #3 
Newbie Joined: Jul 2012 Posts: 3 Thanks: 0  Re: Law of Large Numbers and the Expected Value
So, basically, the LLN says that E(A) will converge to a r.v. X which will assume value u (mean) with probability 1 when the number of trials (samples) goes to infinity.

July 18th, 2012, 05:57 AM  #4 
Newbie Joined: Jul 2012 Posts: 3 Thanks: 0  Re: Law of Large Numbers and the Expected Value
Sorry, I made a mistake. I meant: the LLN says that A will converge to a r.v. X which will assume value u (mean) with probability 1 when the number of trials (samples) goes to infinity. 
July 18th, 2012, 12:23 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,276 Thanks: 516  Re: Law of Large Numbers and the Expected Value
It is unnecessary to call X a random variable, since it has a specific value. Otherwise you are right. http://en.wikipedia.org/wiki/Law_of_large_numbers Above is a thorough discussion, including the precise definitions for the weak and strong laws. 

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