Generating function

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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

 

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