Central Limit Theorem

Let us briefly review how we obtained the Gaussian distribution function in the first place. We started from a very simple system with only two possible outcomes. Of course, the probability distribution function (for ) for this system did not look anything like a Gaussian. However, when we combined very many of these simple systems together, to produce a complicated system with a great number of possible outcomes, we found that the resultant probability distribution function (for ) reduced to a Gaussian in the limit that the number of simple systems tended to infinity. We started from a two outcome system because it was easy to calculate the final probability distribution function when a finite number of such systems were combined together. Clearly, if we had started from a more complicated system then this calculation would have been far more difficult.

Suppose that we start from a general system,
with a general probability distribution function (for some measurable quantity
). It turns out that if
we combine a sufficiently large number of
such systems together then the resultant distribution function
(for
) is always Gaussian.
This astonishing result is known as the *central limit theorem*.
Unfortunately, the central limit theorem is notoriously difficult to prove.
A somewhat restricted proof is presented
in Sections 1.10 and 1.11 of Reif.
The central limit theorem guarantees that the probability distribution of
any measurable quantity
is Gaussian, provided that a sufficiently large number
of statistically independent observations are made. We can, therefore,
confidently predict that Gaussian probability distributions are going to crop up
very frequently in
statistical thermodynamics.