TITLE: Why do bell curves appear everywhere?
QUESTION [95 upvotes]: Why do most probability graphs show a bell curve? I've been wondering why... Is it just something natural, like the fibonacci sequence?

REPLY [2 votes]: Heat is random motion. The rate of flow of random motion behaves like the flow of heat.
The rate of change of heat at a position is the rate of heat going in minus the heat going out. The rate of heat transfer is proportional to the difference in temperature. In one dimension,
$$\frac{du}{dt} = k \frac{d^2u}{dx^2}$$
Suppose you have a spike of heat. All the heat in one place.
$$u(x,0)=\delta(x)$$
where $\delta$ is the Kronecker delta function.
With $\sigma^2 = kt$ and $\mu=0$, the normal (Gaussian) distribution is a solution to this initial value problem.
$$\Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right)$$
Over time the heat will fall down into a bell curve. This gives an intuitive idea of how randomness flows away from an initial exact position towards the two tails of the bell curve.