# Generating function

Pocket

The Generating function of a probability distribution $f(k)$ is given as follows :

$G[x]=\sum_{k=0}^\infty f(k)x^k$

When this Generating function becomes an elementary function, it is very useful to calculate in a simple manner a variety of metrics such as expectation value or variance. However, when this Generating function is not enough simple (e.g. the generating function of power-law), the calculation will become more complex and it may not have a clear benefit.

When the Generating function is enough simple (e.g. Poisson distribution), the computation of expectation value and variance is very simple by using the following formula:

$E=\frac{dG(x)}{dx}|_{x=1}$

$V=\frac{d^2G(x)}{dx^2}|_{x=1}+\frac{dG(x)}{dx}|_{x=1}-E^2$