Exponential distribution information

Exponential
	Probability density function
	Cumulative distribution function
Parameters	rate, or inverse scale
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF
CF
Fisher information
Kullback–Leibler divergence
Expected shortfall

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.

Exponential distribution information

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Exponential distribution

Double exponential distribution

Exponential family

List of probability distributions

Wrapped exponential distribution

Laplace distribution

Gamma distribution

Exponentially modified Gaussian distribution

Erlang distribution

Geometric distribution

Pareto distribution

Weibull distribution

Gumbel distribution

Subexponential distribution

Normal distribution

Kaniadakis exponential distribution

Order statistic

Hypoexponential distribution

Lomax distribution

Variance

Exponential

Bernoulli distribution

Rayleigh distribution

Stretched exponential function

Beta distribution

Exponential
Probability density function
Cumulative distribution function
Parameters	$\lambda >0,$ rate, or inverse scale
Support	$x\in [0,\infty )$
PDF	$\lambda e^{-\lambda x}$
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	$2$
Excess 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	$\ln {\frac {\lambda _{0}}{\lambda }}+{\frac {\lambda }{\lambda _{0}}}-1$
Expected shortfall	$-{\frac {\ln(1-p)+1}{\lambda }}$