Exponential distribution

In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Exponential

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \lambda >0,}$ rate, or inverse scale
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle \lambda e^{-\lambda x}}$
CDF	${\displaystyle 1-e^{-\lambda x}}$
Quantile	${\displaystyle -{\frac {\ln(1-p)}{\lambda }}}$
Mean	${\displaystyle {\frac {1}{\lambda }}}$
Median	${\displaystyle {\frac {\ln 2}{\lambda }}}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle {\frac {1}{\lambda ^{2}}}}$
Skewness	${\displaystyle 2}$
Ex. kurtosis	${\displaystyle 6}$
Entropy	${\displaystyle 1-\ln \lambda }$
MGF	${\displaystyle {\frac {\lambda }{\lambda -t}},{\text{ for }}t<\lambda }$
CF	${\displaystyle {\frac {\lambda }{\lambda -it}}}$
Fisher information	${\displaystyle {\frac {1}{\lambda ^{2}}}}$
Kullback–Leibler divergence	${\displaystyle \ln {\frac {\lambda _{0}}{\lambda }}+{\frac {\lambda }{\lambda _{0}}}-1}$
Expected shortfall	${\displaystyle -{\frac {\ln(1-p)+1}{\lambda }}}$

The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions.