The Generating function of a probability distribution is given as follows :
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: