
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
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: 6,341 Thanks: 532  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: 6,341 Thanks: 532  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. 

Tags 
expected, large, law, numbers 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Dividing Very Large Numbers  nogar  Elementary Math  1  March 14th, 2014 03:39 PM 
Law of Large Numbers and Expected Value  Tomi  Advanced Statistics  0  February 12th, 2014 12:00 PM 
The last two digits of LARGE numbers?  ricsi046  Number Theory  2  November 10th, 2013 05:31 AM 
Ratio and large numbers/powers  alexpasty  Elementary Math  7  March 9th, 2013 01:32 PM 
Weak Law of Large Numbers  Artus  Advanced Statistics  0  January 29th, 2013 12:14 AM 