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,
![\[ f\left (x;a,\theta \right )=\frac{a_{x}\theta ^{x}}{f(\theta )}; x=0,1,2,3,...;a_{x}\geq 0 \]](https://www.statisticalaid.com/wp-content/ql-cache/quicklatex.com-dae6dccb63ca16c809c43c0cbf592e17_l3.png "Rendered by QuickLaTeX.com")
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) = eθ 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, …}, |