**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.

I like the concise nature of your mathematical calculation breakdowns in explaining the nature of the theorems and the statistical analysis tools you cover. Thank you for your information!

Jim,

Thank you for your comment. I do my best to make the information accessible.

Charles