**Central Limit Theorem**: If *x* has a distribution with mean *μ* and standard deviation *σ* then for *n* sufficiently large, the variable

has a distribution which is approximately the standard normal distribution.

Proof: Using Properties 3 and 4 of General Properties of Distributions, and the fact that all the *x _{i}* are independent with the same distribution, we have

Taking the natural log of both sides of the equation, we get

As we can see from the proof of Property 2 of General Properties of Distributions,

If we set *z* = all the terms on the right after 1 +, we have

Thus for sufficiently large *n*, |*z*| < 1. It then follows that

where *q* consists of all the terms in *θ ^{k}* in the double expansion with

*k*≥ 3. Note too that the only function of

*n*that each of these terms have is of the form

*n*. Rearranging the terms, we have

^{– k ⁄ 2}Since and , we conclude that

But as *n* → ∞ we see that *nqθ*^{3} → 0. Thus

By Property 3 of Normal Distribution, a random variable with standard normal distribution has the same moment generating function, and so the result follows from Corollary 1 of General Properties of Distributions.