Editing Exponential distribution (section)

{{pp|small=yes}}
{{Short description|Probability distribution}}
{{Distinguish|text=the [[exponential family]] of probability distributions}}
{{Probability distribution
  | name       = Exponential
  | type       = continuous
  | pdf_image  = [[File:Exponential distribution pdf - public domain.svg|325px|plot of the probability density function of the exponential distribution]]
  | cdf_image  = [[File:Exponential distribution cdf - public domain.svg|325px|Cumulative distribution function]]
  | parameters = <math>\lambda > 0,</math> rate, or inverse [[scale parameter|scale]]
  | support    = <math>x \in [0, \infty)</math>
  | pdf        = <math>\lambda e^{-\lambda x}</math>
  | cdf        = <math>1 - e^{-\lambda x}</math>
  | quantile   = <math>-\frac{\ln(1 - p)}{\lambda}</math>
  | mean       = <math>\frac{1}{\lambda}</math>
  | median     = <math>\frac{\ln 2}{\lambda}</math>
  | mode       = <math>0</math>
  | variance   = <math> \frac{1}{\lambda^2}</math>
  | skewness   = <math>2</math>
  | kurtosis   = <math>6</math>
  | entropy    = <math>1 - \ln\lambda</math>
  | mgf        = <math>\frac{\lambda}{\lambda-t}, \text{ for } t < \lambda</math>
  | char       = <math>\frac{\lambda}{\lambda-it}</math>
  | fisher     = <math>\frac{1}{\lambda^2}</math>
  | KLDiv      = <math>\ln\frac{\lambda_0}{\lambda} + \frac{\lambda}{\lambda_0} - 1</math>
  |ES=<math>\frac{-\ln(1 - p) + 1}{\lambda}</math>|bPOE=<math>e^{1-\lambda x}</math>}}

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.<ref>{{Cite web |date=2021-07-15 |title=7.2: Exponential Distribution |url=https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Inferential_Statistics_and_Probability_-_A_Holistic_Approach_(Geraghty)/07:_Continuous_Random_Variables/7.02:_Exponential_Distribution |access-date=2024-10-11 |website=Statistics LibreTexts |language=en}}</ref> 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]].<ref>{{Cite web |title=Exponential distribution {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/exponential-distribution |access-date=2024-10-11 |website=www.britannica.com |language=en}}</ref> In addition to being used for the analysis of Poisson point processes it is found in various other contexts.<ref name="Weisstein">{{Cite web |last=Weisstein |first=Eric W. |title=Exponential Distribution |url=https://mathworld.wolfram.com/ExponentialDistribution.html |access-date=2024-10-11 |website=mathworld.wolfram.com |language=en}}</ref>

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 distribution|normal]], [[binomial distribution|binomial]], [[gamma distribution|gamma]], and [[Poisson distribution|Poisson]] distributions.<ref name="Weisstein" />