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	${\displaystyle 2}$
Excess 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	${\displaystyle -{\frac {\ln(1-p)+1}{\lambda }}}$

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.

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.

Definitions

Probability density function

The probability density function (pdf) of an exponential distribution is

${\displaystyle f(x;\lambda )={\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}$

Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ Exp(λ).

The exponential distribution exhibits infinite divisibility.

Cumulative distribution function

The cumulative distribution function is given by

${\displaystyle F(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}$

Alternative parametrization

The exponential distribution is sometimes parametrized in terms of the scale parameter β = 1/λ, which is also the mean:

Properties

Mean, variance, moments, and median

The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by

Zdroj:https://en.wikipedia.org?pojem=Exponential_distribution
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