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,

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