The central limit theorem states that the sum of N iid random variables will look more and more like a bell curve when even if s themselves are not normally distributed.

Demonstrating central limit theorem using N numbers of dice

Central Limit Theorem

under the assumptions that

  1. Each is iid (independent and from the same distribution)
  2. The variance is finite 1

As a result of the central limit theorem, we can approximate discrete distributions as a normal distribution.

Intuition

An intuitive explanation of the central limit theorem is that sum of random variables can be viewed as a sequence of convolution operations. As convolution smooths functions, repeatedly applying convolutions eventually gives us the bell-curve shape.

convolution smooths functions

References

Footnotes

  1. In certain continuous distributions (e.g. a t-distribution with 2 DOF), the variance can diverge into infinity