Power Series Distribution definition, formula with applications

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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

 

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