# Exponential distribution

## Probability distribution / From Wikipedia, the free encyclopedia

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In probability theory and statistics, the **exponential distribution** or **negative exponential distribution** is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. 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.

**Quick facts: Parameters, Support, PDF, CDF, Quantile...**▼

Probability density function | |||

Cumulative distribution function | |||

Parameters | $\lambda >0,$ rate, or inverse scale | ||
---|---|---|---|

Support | $x\in [0,\infty )$ | ||

$\lambda e^{-\lambda x}$ | |||

CDF | $1-e^{-\lambda x}$ | ||

Quantile | $-{\frac {\ln(1-p)}{\lambda }}$ | ||

Mean | ${\frac {1}{\lambda }}$ | ||

Median | ${\frac {\ln 2}{\lambda }}$ | ||

Mode | $0$ | ||

Variance | ${\frac {1}{\lambda ^{2}}}$ | ||

Skewness |
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| ||

Ex. kurtosis | $6$ | ||

Entropy | $1-\ln \lambda$ | ||

MGF | ${\frac {\lambda }{\lambda -t}},{\text{ for }}t<\lambda$ | ||

CF | ${\frac {\lambda }{\lambda -it}}$ | ||

Fisher information | ${\frac {1}{\lambda ^{2}}}$ | ||

Kullback–Leibler divergence | $\ln {\frac {\lambda _{0}}{\lambda }}+{\frac {\lambda }{\lambda _{0}}}-1$ |

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.