The exponential distribution

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The exponential distribution describes the time between events if these events follow a Poisson process (i.e. a stochastic process in which events occur continuously and independently of one another). It is also called the negative exponential distribution. It is not the same as the exponential family of distributions.




Distribution name

Exponential distribution

Common notation


 = inverse scale (i.e. rate) parameter ()


Probability density function

Cumulative distribution function




(Excess) kurtosis

Characteristic function

Other comments

The exponential distribution is a special case of the Gamma distribution, as if  then .


The mode of an exponential distribution is 0. The quantile function, i.e. the inverse cumulative distribution function, is .


The non-central moments ( are . Its median is .


Nematrian web functions


Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “exponential”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “exponential2” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.


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