Proof of Bernoulli's LawDate: 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/ |
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