Proof of Bernoulli's Law

Date: 6/22/96 at 15:29:33
From: peter
Subject: Probability - Bernoulli's law of large numbers

I am a high school student from Sweden and I would like to know how 
one can prove Bernoulli's law of large numbers, which states that if 
an experiment is repeated n times, the relative frequencey of an 
occurrence tends to a fixed number "p" as n tends to infinity. 

This theorem seems to me to be more of an empirical fact than a 
derived theorem of axiomatic math, so I really wonder...

Thank you.

Date: 6/23/96 at 20:42:20
From: Doctor Anthony
Subject: Re: Probability - Bernoulli's law of large numbers

Explanation of Law of Large Numbers.

Let X(r) be a sequence of mutually independent random variables with a 
common distribution. If the expectation MU = E(X(r)) exists then:

PR[{(X(1)+X(2)+..+X(n)}/n - MU > k] -> 0  as n -> infinity for any k>0

If S(n) = X(1)+X(2)+....+X(n)  and if the variance is VAR(X) then by 
Chebyshev's inequality:

  PR[S(n)>t] <or= nVAR(X)/t^2

For t = kn the right hand side tends to 0 and so:

  PR[S(n)>kn] -> 0

  PR[S(n)/n > k] -> 0

  PR[S(n)/n - MU > k] -> 0

Which is the condition we wished to prove.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/