Mean = Most Probable Value - Statistical Mechanics

[Sekonda]

Hey guys,

In statistical mechanics I need to explain why the mean value is approximately equal to the most probable value for systems with a large number of random variables.

Now I can provide an example of the binomial distribution and show what happens when N tends to infinity ( it goes to a dirac delta function), however I'm not sure how to explain why this happens exactly.

I believe the basis is the Law of Large numbers, I'm not sure if stating this law is sufficient enough to though.

Cheers for any help,
Tom

 

[jbsteele]

Yes, the Law of Large Numbers is a good starting point to explain why the mean value is approximately equal to the most probable value for systems with a large number of random variables. The Law of Large Numbers states that as the number of independent and identically distributed random variables increases, the sample mean converges to the expected value (or mean) of the underlying population distribution. This means that as the number of random variables (N) increases, the probability of any given value in the distribution becomes more and more concentrated around the mean. In the case of a binomial distribution, as N tends to infinity, the probability of any single value tends to zero, and the only value that has any probability of occurring is the mean value.