My Math Forum Law of Large Numbers and the Expected Value

 July 17th, 2012, 01:01 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, 01:33 PM #2 Global Moderator   Joined: May 2007 Posts: 6,399 Thanks: 546 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, 06:49 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, 06:57 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, 01:23 PM #5 Global Moderator   Joined: May 2007 Posts: 6,399 Thanks: 546 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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