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.