# Exponentiated exponential distribution

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

The cumulative distribution function (CDF) of the exponentiated exponential distribution is given by [2] (for α, λ > 0):

where

The distribution has an increasing or decreasing failure rate depending on the value of the shape parameter.

The probability density function (PDF) is [2]

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

## Exponential vs. exponentiated exponential distribution

While both the Exponential distribution and the exponentiated Exponential distribution have a similar form for their probability density function, they are different distributions with different properties.

The Exponential distribution has a single parameter, which is the rate parameter λ. The probability density function (PDF) of the Exponential distribution is given by f(x) = λ * exp(-λx), where x >= 0 and λ > 0.

The Exponential distribution is used to model the time between events in a Poisson process. It has a memoryless property, which means that the probability of an event occurring within a certain time interval is not affected by how much time has already passed.

Overall, the main difference between the two distributions is that the Exponential distribution has a single parameter (rate parameter), while the exponentiated Exponential distribution has two parameters (shape and scale parameters).

## References

[1] Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299-302.

[2] R. D. Gupta, D. Kundu: Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distribution. BiometricalJournal 43 (2001) 1, 117–130

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