My Math Forum  

Go Back   My Math Forum > College Math Forum > Advanced Statistics

Advanced Statistics Advanced Probability and Statistics Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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?
jimmy is offline  
 
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.
mathman is offline  
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.
jimmy is offline  
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.
jimmy is offline  
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.
mathman is offline  
Reply

  My Math Forum > College Math Forum > Advanced Statistics

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 04:39 PM
Law of Large Numbers and Expected Value Tomi Advanced Statistics 0 February 12th, 2014 01:00 PM
The last two digits of LARGE numbers? ricsi046 Number Theory 2 November 10th, 2013 06:31 AM
Ratio and large numbers/powers alexpasty Elementary Math 7 March 9th, 2013 02:32 PM
Weak Law of Large Numbers Artus Advanced Statistics 0 January 29th, 2013 01:14 AM





Copyright © 2017 My Math Forum. All rights reserved.