The Kaniadakis exponential distribution (or κ-exponential distribution) is a probability distribution arising from the maximization of the Kaniadakis entropy under appropriate constraints. It is one example of a Kaniadakis distribution. The κ-exponential is a generalization of the exponential distribution in the same way that Kaniadakis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The κ-exponential distribution of Type I is a particular case of the κ-Gamma distribution, whilst the κ-exponential distribution of Type II is a particular case of the κ-Weibull distribution.

The Kaniadakis κ-exponential distribution of Type I is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics which exhibit power-law tails. This distribution has the following probability density function:

valid for ${\displaystyle x\geq 0}$, where ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index associated with the Kaniadakis entropy and ${\displaystyle \beta >0}$ is known as rate parameter. The exponential distribution is recovered as ${\displaystyle \kappa \rightarrow 0.}$

The cumulative distribution function of κ-exponential distribution of Type I is given by

for ${\displaystyle x\geq 0}$. The cumulative exponential distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

The κ-exponential distribution of type I has moment of order ${\displaystyle m\in \mathbb {N} }$ given by

where ${\displaystyle f_{\kappa }(x)}$ is finite if ${\displaystyle 0<m+1<1/\kappa }$.

The expectation is defined as: