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The **power function distribution** (PFD) is a flexible model used to analyze and model income distribution data, lifetime data, and modeling of failure processes. One strength of the power function distribution is its mathematical simplicity, compared to more complex distributions like the Weibull distribution.

The power function distribution is a special case of the beta distribution [1], and the Pearson type I distribution. It also has an inverse relationship with the standard Pareto distribution [2]; moments of the PFD are the negative moments of the Pareto distribution [3].

Sometimes the graph of a power function of the form f(x) = ax^{p } is called a *power function distribution.* In addition, there are various different versions of the PDFs and CDFs.

## Power Function Distribution Formulas

A random variable has a two-parameter PFD if its Probability Density Function (PDF) is [4]:

With Cumulative distribution function of

There are many variations in the literature, including this PDF given by [5]

With CDF

This three-parameter power function distribution has a threshold parameter θ, scale parameter σ, and shape parameter α. The density function for

*θ* < *x* < *θ* + *σ*

is given as [6]

**References**

[1] Pandey, A. & Saran, J. (2004). ESTIMATION OF PARAMETERS OF A POWER FUNCTION DISTRIBUTION AND ITS CHARACTERIZATION BY K-TH RECORD VALUES. STATISTICA, anno LXIV, n.3. Online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.953.5181&rep=rep1&type=pdf

[2] Kleiber C, Kotz S. Statistical size distributions in economics and actuarial sciences: John Wiley & Sons; 2003.

[3] Johnson NL, Kotz S. Distributions in Statistics: Continuous Univariate Distributions: Vol 1. John Wiley & Sons; 1970.

[4] Ansari, S. et al. (2019). Cubic Transmuted PFD. Gazi University Journal of Science. 32(4): 1322-1337. Online: https://scholar.ppu.edu/bitstream/handle/123456789/2026/2019_Cubic%20Transmuted%20Power%20Function%20Distribution.pdf?sequence=1&isAllowed=y

[5] Saran & Pandey. (2004). ESTIMATION OF PARAMETERS OF A POWER FUNCTION DISTRIBUTION AND ITS CHARACTERIZATION BY K-TH RECORD VALUES. STATISTICA, anno LXIV, n. 3

[6] Wicklin, R. (2003). Simulating Data with SAS. SAS Institute.