### 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(Î¸) is a generating function and f(Î¸) 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 Î¸=p/(1-p), f(Î¸)=(1+Î¸)^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(Î¸)=e^Î¸ and s={0,1,2,3,...,∞} then the distribution tends to poisson distribution.
- If Î¸=p/(1-p), f(Î¸)=(1+Î¸)^-n and s={0,1,2,3,...,∞), then the power series distribution tends to negative binomial distribution.
- If f(Î¸)=-log(1-Î¸) and s={1,2,....}, then the power series distribution tends to logarthmic distribution.

Characteristics of power series distribution

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