# ExponentialDistribution

represents an exponential distribution with scale inversely proportional to parameter λ.

# Details • The probability density for value in an exponential distribution is proportional to for , and is zero for . »
• ExponentialDistribution allows λ to be any positive real number.
• ExponentialDistribution allows λ to be a quantity of any unit dimension. »
• ExponentialDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

# Background & Context

• represents a continuous statistical distribution defined over the interval and parametrized by a positive real number λ. The probability density function (PDF) of an exponential distribution is monotonically decreasing. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The exponential distribution is sometimes referred to as the negative exponential distribution, the one-parameter exponential distribution, or the antilogarithmic distribution.
• Historically, the exponential distribution has been used most widely to describe events recurring "at random in time", i.e. in circumstances for which the future lifetime of an individual has the same distribution regardless of its present state. Use of the exponential distribution has increased significantly over the last 75 years, due in part to considerable research within the field of order statistics beginning in the early to mid-1950s. Since then, the exponential distribution has been used to model various phenomena over intervals of approximately constant rate, e.g. the number of phone calls placed in a specific time interval each day. In stochastic processes, the exponential distribution describes the lengths of interarrival times in homogeneous Poisson processes (i.e. continuous-time counting processes whose increments are independent, stationary, and Poisson-distributedimplemented as PoissonProcess). The exponential distribution is also used in credit risk modeling, queueing theory, reliability theory, physics, and hydrology.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an exponential distribution. Distributed[x,ExponentialDistribution[λ]], written more concisely as xExponentialDistribution[λ], can be used to assert that a random variable x is distributed according to an exponential distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[ExponentialDistribution[λ],x] and CDF[ExponentialDistribution[λ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with an exponential distribution, EstimatedDistribution to estimate an exponential parametric distribution from given data, and FindDistributionParameters to fit data to an exponential distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic exponential distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic exponential distribution.
• TransformedDistribution can be used to represent a transformed exponential distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an exponential distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving exponential distributions.
• The exponential distribution is related to a large number of other distributions. Its occurrence relative to a PoissonProcess induces a relationship with PoissonDistribution and CompoundPoissonDistribution. ExponentialDistribution can be thought of as a basis for the extreme value distributions family, due to the fact that can be transformed into each of ExtremeValueDistribution, GumbelDistribution, FrechetDistribution, and WeibullDistribution, while ExponentialDistribution can be obtained from each of GammaDistribution, LaplaceDistribution, BenktanderWeibullDistribution, LogisticDistribution, ParetoDistribution, PearsonDistribution, PowerDistribution, and RayleighDistribution by way of TransformedDistribution and/or TruncatedDistribution. Several distributions can be derived by combining various other distributions with ExponentialDistribution, e.g. GeometricDistribution (by combining ExponentialDistribution with PoissonDistribution), KDistribution (by combining with GammaDistribution), HoytDistribution (by combining with ArcSinDistribution), and ParetoDistribution (by combining with either ErlangDistribution or GammaDistribution).

# Examples

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## Basic Examples(4)

Probability density function:

 In:= Out= In:= Out= Cumulative distribution function:

 In:= Out= In:= Out= Mean and variance:

 In:= Out= In:= Out= Median:

 In:= Out= ## Neat Examples(1)

Introduced in 2007
(6.0)
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Updated in 2016
(10.4)