A power series distribution is a discrete probability distribution that applies to a subset of natural numbers. The distribution’s names derive from the fact that the power series is used to construct them. This broad category of distributions includes a number of discrete distributions, such as the Poisson distribution, negative binomial distribution, and binomial distribution.

### Power Series Distribution

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

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

### Properties

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

Power Series Distribution to others distribution…

Power Series Tends to- | Condition |

Binomial distribution | If θ = p / (1 – p); f(θ) = (1 + θ)^{n}; s = {1, 2, 3, … n} |

Poisson Distribution | If f(θ) = e^{θ} and s = {0, 1, 2, 3, … ∞} |

Negative Binomial Distribution | If θ = p / (1 – p); f(θ) = (1 + θ)^{-n}; s = {0, 1, 2, 3, … ∞} |

Logarithmic distribution | If f(θ) = -log (1 – θ) and s = {1, 2, …}, |