# Power Series Distribution definition, formula with applications

### Power Series Distribution

A discrete random variable X is said to have a generalised power series distribution if its probability function is given by,

$f\left&space;(&space;x;a,\theta&space;\right&space;)=\frac{a_{x}\theta&space;^{x}}{f\left&space;(&space;\theta&space;\right&space;)};&space;x=0,1,2,3....;&space;a_{x}\geq&space;0.$

where f(θ) is a generating function and  f(θ) is positive finite and diferentiable. Power series distribution is a discrete probability distribution.

### Properties of Power series distribution

Some special properties of power series distribution are given-
• If θ=p/(1-p), f(θ)=(1+θ)^n and s={1,2,3,...,n), a set of (n+1) non-negative integers then the power series distribution is tends to binomial distribution.
• If f(θ)=e^θ and s={0,1,2,3,...,∞} then the distribution tends to poisson distribution.
• If θ=p/(1-p), f(θ)=(1+θ)^-n and s={0,1,2,3,...,∞), then the power series distribution tends to negative binomial distribution.
• If f(θ)=-log(1-θ) and s={1,2,....}, then the power series distribution tends to logarthmic distribution.
Characteristics of power series distribution