The Exponential distribution arises as the distribution of waiting times (intervals between successive events) in a Poisson process with constant rate, λ. If events are occurring at random at a constant rate, λ, per unit time and x is the time to the first occurrence, then the most commonly used form of the Exponential distribution is:
or alternatively:
Note that for x=0 f(x)=1/β.
The cdf is:
The simplest form of the distribution is where λ=1, giving the standard Exponential distribution:
The relationship to the Poisson can be obtained as follows. The Poisson distribution gives the probability that a number of events, x, will be observed based on an intensity parameter, λ:
Consider the case in which there are 0 events in the interval t units of time, and 1 event at the end of this period, i.e. in the time interval [t,t+dt]. From the Poisson, with independent events and a rate per unit time of λ, we expect an average of λt events in t units and λdt events in dt units of time, hence we have:
This result can be extended to two or more dimensions, and to the time (or distance) to the second event (second nearest point in space), which has a number of applications in ecology (e.g. "how far is the nearest tree/bird's nest/patch of insect eggs ...") and in spatial analysis generally (see further, de Smith et al., section 5.4 [DES1]). The distribution exhibits the lack of memory property, i.e. p(X>a+b|X>b)=p(X>a) and is the only continuous pdf with this feature. An Exponential distribution that is truncated from below remains an Exponential distribution.
A more general version of the distribution, with the minimal value of x shifted to the right from 0, is also sometimes used. This has the form:
The Exponential distribution is widely used, both as a convenient and simple general-purpose model and as a representation of the lifetime of a product, such as a light bulb, an electronic component or a mechanical device. In this context the intensity parameter, λ, is considered as the failure rate, and the mean of the distribution, which is 1/λ, is the Mean Time To Failure (MTTF). If the exponential does not provide a suitable fit to lifetime data an alternative model may be used (for a fuller discussion of this question see the NIST/Engineering Handbook), with the Weibull and the Gamma distributions being common alternatives. If the variable Y=(X-θ)c has an Exponential distribution then X has a Weibull distribution, hence the Exponential can be seen as a special case of a Weibull. Sample plots of the distribution, using the MATLab exppdf() function with origin at zero for varying values of β=1/λ are shown below:
Exponential distributions, β=1/λ=0.5, 0.75, 1, 1.5 and 2
Key measures for the Exponential distribution are provided in the table below:
Item |
Value |
|---|---|
Mean |
1/λ |
Variance |
1/λ2 |
Skewness |
2 |
Kurtosis |
6 |
MGF |
λ/(λ-t), t<λ |
References
[DES1] de Smith M J, Goodchild M F, Longley P A (2018) Geospatial Analysis: A Comprehensive Guide to Principles, Techniques and Software Tools. 6th edition, The Winchelsea Press, UK. Available from: https://www.spatialanalysisonline.com/
[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York
Mathworld: Weisstein E W: Exponential distribution: https://mathworld.wolfram.com/ExponentialDistribution.html
NIST Engineering Statistics Handbook: Lifetime distribution models: https://www.itl.nist.gov/div898/handbook/apr/section1/apr16.htm
Wikipedia: Exponential distribution: https://en.wikipedia.org/wiki/Exponential_distribution