< List of probability distributions > *Exponentiated exponential distribution*

## What is the exponentiated exponential distribution?

The **exponentiated exponential distribution** is a continuous probability distribution used to model a variety of phenomena in various fields, including economics, engineering and medicine. It is a generalization of the exponential distribution; It is also a particular case of the exponentiated Weibull distribution [1] and the beta exponential distribution.

## Properties

The exponentiated exponential distribution cumulative distribution function (CDF) is [2]:

(for α, λ > 0), where

- α = shape parameter (this value can lead to increasing or decreasing failure rates).
- λ = scale parameter.

The probability density function (pdf) is [2]

When α = 1, the distribution reduces to an exponential distribution.

## Exponential vs. exponentiated exponential distribution

Both the exponential and exponentiated exponential distribution have similar looking pdfs; The pdf of an exponential distribution is given by

**f(x) = λ * exp(-λx),**

where x >= 0 and λ > 0.

However, the two distributions have different properties. For example, while the exponentiated exponential has two parameters, the exponential distribution has just one — the rate parameter λ.

## References

- Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299-302.
- R. D. Gupta, D. Kundu: Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distribution. BiometricalJournal 43 (2001) 1, 117–130