TITLE: Intuition about the Central Limit Theorem
QUESTION [17 upvotes]: I'm studying statistics, and would like to better understand the Central Limit Theorem. The proof I found on Wikipedia requires some previous knowledge I do not currently possess.
Is there a quick intuitive explanation you can give as to why this theorem is correct?

REPLY [2 votes]: This answer gives an outline of how to use the Fourier Transform to prove that the $n$-fold convolution of any probability distribution with a finite variance contracted by a factor of $\sqrt{n}$ converges weakly to the normal distribution.
However, in his answer, Qiaochu Yuan mentions that one can use the Principle of Maximum Entropy to get a normal distribution. Below, I have endeavored to do just that using the Calculus of Variations.

Applying the Principle of Maximum Entropy
Suppose we want to maximize the entropy
$$
-\int_{\mathbb{R}}\log(f(x))f(x)\,\mathrm{d}x\tag1
$$
over all $f$ whose mean is $0$ and variance is $\sigma^2$, that is
$$
\int_{\mathbb{R}}\left(1,x,x^2\right)f(x)\,\mathrm{d}x=\left(1,0,\sigma^2\right)\tag2
$$
That is, we want the variation of $(1)$ to vanish
$$
\int_{\mathbb{R}}(1+\log(f(x)))\,\delta f(x)\,\mathrm{d}x=0\tag3
$$
for all variations of $f$, $\delta f(x)$, so that the variation of $(2)$ vanishes
$$
\int_{\mathbb{R}}\left(1,x,x^2\right)\delta f(x)\,\mathrm{d}x=(0,0,0)\tag4
$$
$(3)$, $(4)$, and orthogonality requires
$$
\log(f(x))=c_0+c_1x+c_2x^2\tag5
$$
To satisfy $(2)$, we need $c_0=-\frac12\log\left(2\pi\sigma^2\right)$, $c_1=0$, and $c_2=-\frac1{2\sigma^2}$. That is,
$$
\bbox[5px,border:2px solid #C0A000]{f(x)=\frac1{\sigma\sqrt{2\pi}}\,e^{-\frac{x^2}{2\sigma^2}}}\tag6
$$