< List of probability distributions

A power series distribution is a discrete probability distribution on a subset of natural numbers constructed from the power series. A power series is an infinite series of the form

Many distributions, such as the binomial distribution, negative binomial distribution, and Poisson distribution, are classified into this general category.

## Properties of a power series distribution

A discrete random variable *X* has the power series distribution if its probability density function (PDF) is

Where f(θ) is a positive, finite, and differentiable generating function [1]. when θ = 0, the distribution is the point mass distribution at 0 (i.e., f_{(0)} = 1) [2].

The generalized power series distribution belongs to the exponential family and can be expressed as follows [3].

where

- Where
*a*and*g*are functions of an unknown parameter*θ*, *c*is a function of*x.*

The power series distribution has a few special properties:

Condition | Power Series Tends to… |

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

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

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

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

## Power Series Distribution Variants

The literature describes several compounded distributions such as the Weibull-power series distribution and the generalized Gompertz-power series distributions. These are obtained by compounding the generalized Gompertz distribution (a generalization of the exponential distribution) and the power series distribution [4].

In addition, there is an exponential power series distribution that is a composition of the exponential distribution with the power series distribution. This gives a distribution with a decreasing failure rate [5].

## Power series vs power law distributions

A power series distribution is a probability distribution defined by an infinite series of terms, often of the form x^{n}, where n is a non-negative integer, and commonly used for phenomena that exhibit a long tail—a few large values and many small ones.

On the other hand, a power law distribution is defined by a power law function, f(x) = x − α, where α is a positive constant. It is often used to model scale-free phenomena whose values do not change when scaled by a constant factor.

One significant difference between these types of distributions is that the power series distribution has a finite mean and variance, while the power law distribution does not. The following table summarizes the differences between the two distributions:

Property | Power Series Distribution | Power Law Distribution |
---|---|---|

Number of terms | Infinite | Finite |

Function | Power series | Power law |

Mean | Finite | Undefined |

Variance | Finite | Undefined |

Long tail | Yes | Yes |

Scale-free | No | Yes |

*Table showing the major differences between power law and power series distributions.*

## References

[1] Gupta, R. (1974). Modified Power Series Distribution and Some of Its Applications. The Indian Journal of Statistics, Volume 36, Series B, Pt. 3, pp. 288-298.

[2] Power series distributions. Retrieved May 20, 2023 from: https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.05%3A_Power_Series_Distributions

[3] Sakia, R. (2018). Application of the Power Series Probability Distributions for the Analysis of Zero-Inflated Insect Count Data. Open Access Library Journal, Vol.5 No.10.

[4] Tahmasebi, S. & Jafari, A. (2000). Generalized Gompertz-Power Series Distributions. Retrieved December 14, 2021 from: https://www.academia.edu/25858978/Generalized_Gompertz-Power_Series_Distributions

[5] Chahkandi, M. & Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53(12): 4433-4440.