Exponential distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Not to be confused with the exponential family of probability distributions.
Probability density function
Probability density function
Cumulative distribution function
Cumulative distribution function
Parameters λ > 0 rate, or inverse scale
Support x ∈ [0, ∞)
PDF λ eλx
CDF 1 − eλx
Quantile −ln(1 − F) / λ
Mean λ−1 (= β)
Median λ−1 ln(2)
Mode 0
Variance λ−2 (= β2)
Skewness 2
Ex. kurtosis 6
Entropy 1 − ln(λ)
Fisher information




Probability density function[edit]

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

Alternatively, this can be defined using the right-continuous Heaviside step function, H(x) where H(0)=1:


The exponential distribution exhibits infinite divisibility.

Cumulative distribution function[edit]

The cumulative distribution function is given by

Alternatively, this can be defined using the Heaviside step function, H(x).

Alternative parameterization[edit]

A commonly used alternative parametrization is to define the probability density function (pdf) of an exponential distribution as

where β > 0 is mean, standard deviation, and scale parameter of the distribution, the reciprocal of the rate parameter, λ, defined above. In this specification, β is a survival parameter in the sense that if a random variable X is the duration of time that a given biological or mechanical system manages to survive and X ~ Exp(β) then E[X] = β. That is to say, the expected duration of survival of the system is β units of time. The parametrization involving the "rate" parameter arises in the context of events arriving at a rate λ, when the time between events (which might be modeled using an exponential distribution) has a mean of β = λ−1.

The alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. This alternative specification is not used here. Unfortunately this gives rise to a notational ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "X ~ Exp(λ)", since either the notation in the previous (using λ) or the notation in this section (here, using β to avoid confusion) could be intended. An example of this notational switch: reference[1] uses λ for β.


Mean, variance, moments and median[edit]



, see above.


The variance of X is given by


so the standard deviation is equal to the mean.

The moments of X, for n = 1, 2, ..., are given by


The median of X is given by




in accordance with the median-mean inequality.


An exponentially distributed random variable T obeys the relation






Tukey anomaly criteria for exponential probability distribution function.
Tukey criteria for anomalies.[2]


The quartiles are therefore:

  • first quartile: ln(4/3)/λ
  • median: ln(2)/λ
  • third quartile: ln(4)/λ


Kullback–Leibler divergence[edit]

The directed Kullback–Leibler divergence of ('approximating' distribution) from ('true' distribution) is given by

Maximum entropy distribution[edit]


Distribution of the minimum of exponential random variables[edit]


is also exponentially distributed, with parameter


This can be seen by considering the complementary cumulative distribution function:

The index of the variable which achieves the minimum is distributed according to the law

Note that

is not exponentially distributed.[4]

Parameter estimation[edit]


Maximum likelihood[edit]



is the sample mean.

The derivative of the likelihood function's logarithm is:

Consequently, the maximum likelihood estimate for the rate parameter is:

Although this is not an unbiased estimator of , is an unbiased[5] MLE[6] estimator of where is the scale parameter defined in the 'Alternative parameterization' section above and the distribution mean.

Approximate Minimizer of Expected Squared Error[edit]


This is derived from the mean and variance of the Inverse-gamma distribution: .[7]

Confidence intervals[edit]


which is also equal to:

where χ2

This approximation may be acceptable for samples containing at least 15 to 20 elements.[9]

Bayesian inference[edit]





Generating exponential variates[edit]




Other methods for generating exponential variates are discussed by Knuth[10] and Devroye.[11]


Related distributions[edit]

  • Exponential distribution is closed under scaling by a positive factor. If X ~ Exp(λ) then kX ~ Exp(λ/k).
  • If X ~ Exp(1/2) then X ∼ χ2
    , i.e. X has a chi-squared distribution with 2 degrees of freedom. Hence:
  • If Xi ~ Exp(λ) then the sum X1 + ... + Xk = ~ Erlang(k, λ) which is just a Gamma(k, λ−1) (in (k, θ) parametrization) or Gamma(k, λ) (in (α,β) parametrization) with an integer shape parameter k.
  • If X ~ Exp(1) then μ − σ log(X) ~ GEV(μ, σ, 0).
  • If X ~ Exp(λ) then X ~ Gamma(1, λ−1) (in (k, θ) parametrization) or Gamma(1, λ) (in (α, β) parametrization).
  • If X ~ Exp(λ) and Y ~ Exp(ν) then λX − νY ~ Laplace(0, 1).
  • If X, Y ~ Exp(λ) then XY ~ Laplace(0, λ−1).
  • If X ~ Laplace(μ, β−1) then |X − μ| ~ Exp(β).
  • If X ~ Exp(1) then (logistic distribution):
  • If Y|X ~ Poisson(X) where X ~ Exp(λ−1) then (geometric distribution)
  • If X ~ Exp(1) and then (K-distribution)
  • The Hoyt distribution can be obtained from Exponential distribution and Arcsine distribution
  • If X ~ Exp(λ) and Y ~ Erlang(n, λ) then:
  • If X ~ Exp(λ) and then
  • If X ~ SkewLogistic(θ), then log(1 + e−X) ~ Exp(θ).
  • If X ~ Exp(λ) and Y = μ − β log(Xλ) then YGumbel(μ, β).
  • Let X ∼ Exp(λX) and Y ∼ Exp(λY) be independent. Then has probability density function . This can be used to obtain a confidence interval for .
  • Gamma mixture: If λ ~ Gamma(scale=k, shape=θ) and X ~ Exponential(rate=λ) then the marginal distribution of X is Lomax(scale=1/k, shape=θ)

Other related distributions:

Applications of exponential distribution[edit]

Occurrence of events[edit]




  • The time it takes before your next telephone call



Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq[12]









which can be considered as

  • (1) a frequentist confidence distribution, obtained from the distribution of the pivotal quantity ;[14]
  • (2) a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of xn+1 and λ by maximization;[15]
  • (3) an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/λ;
  • (4) the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.[16]



Introduction to the exponential distribution for the novice[edit]

The exponential distribution is used to model the time between the occurrence of events in an interval of time, or the distance between events in space. The exponential distribution may be useful to model events such as

  • The time between goals scored in a World Cup soccer match
  • The time between customer calls to a help center[17]
  • The time between meteors greater than 1 meter diameter striking earth
  • The time between successive failures of a machine
  • The time from diagnosis until death in patients with metastatic cancer
  • The distance between successive breaks in a pipeline

The exponential distribution is an appropriate model if the following conditions are true.

  • X is the time (or distance) between events, with X > 0.
  • The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
  • The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
  • Two events cannot occur at exactly the same instant.

If these conditions are true, then X is an exponential random variable, and the distribution of X is an exponential distribution. If these conditions are not true, then the exponential distribution is not appropriate. Alternative distributions such as the Weibull or gamma may give a better fit to the data, or a semi-parametric model, such as the Cox proportional-hazards model, may be required for statistical analysis.

The graph of an exponential distribution starts on the y-axis at a positive value (called lambda, λ) and decreases to the right.

Graph of exponential distribution PDF with λ = 1

The exponential distribution is specified by the single parameter lambda (λ). Lambda is the event rate, and may have different names in other applications:

  • event rate
  • rate parameter
  • arrival rate
  • death rate
  • failure rate
  • transition rate

Lambda is the number of events per unit time. The graph of the exponential distribution, called the probability density function (PDF), shows the distribution of time (or distance) between events. The PDF is specified in terms of lambda (events per unit time) and x (time).


  • λ is the rate parameter
  • x > 0
  • e is the number 2.71828 …, the base of the natural logs

The density f(x) is 0 for x less than or equal to 0.

Graphs of exponential distribution PDFs with various lambda

The figure shows the PDF for exponential distributions with λ = 1, 2, 3, and 0.5. The line for each distribution meets the y-axis at lambda. Notice in the figure that when lambda (the event rate) is large, the time between events is small. In particular, the mean time between events is given by 1/λ. For λ = 3, the mean time between events is 1/λ = 1/3. For λ = 0.5, the mean time between events is 1/λ = 1/0.5 = 2.

Olkin [18] gives the following example of data modeled using an exponential distribution. The time between successive failures of the air-conditioning system of a particular jet airplane were recorded:

Time between successive failures = 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95

Histogram of AC failure times with fitted exponential curve

The mean time between failures is 59.6. The median time between failures is 22. Because the exponential distribution is skewed right, the median is less than the mean. The figure shows a histogram of the time between failures and a fitted exponential density curve with λ = 1/(mean time to failure) = 1/59.6 = 0.0168.

Cumulative distribution function[edit]

The probability that the time between events is less than a specified time x is given by the cumulative distribution function (CDF):

The figures below show the probability density and cumulative distribution functions for the exponential distribution with lambda = 1.

Graphs of exponential pdf and cdf with lambda = 1

Examples using the CDF to calculate probability[edit]

Example 1. The time between failures in the air-conditioner example was modelled as an exponential distribution with lambda = 0.0168. For λ = 0.0168, the mean time between failures is 1/0.0168 = 59.5 hours. What is the probability that the time until the next failure is less than 100 hours? This question can be answered using the cumulative distribution function. The graph shows the CDF for this example.

Probability of AC failure in less than 100 hours

The graph shows that, using the cumulative distribution function with λ = 0.0168, the value of x = 100 corresponds to a value of y = 0.81. So the probability that the time until the next failure is less than 100 hours is p = 0.81. The same probability can be calculated using the formula for the CDF:

P(time between events is < x) = 1 − e^(−λx)
P(time between events is < 100)=1-e^(-0.0168×100) = 0.8136

Example 2. For an exponential distribution with λ = 4, what is the probability that the time between events (x) is less than 0.5?

From the equation for the CDF, P(time between events is < 0.5) = 1 − e^(-4·0.5) = 0.865

This probability may also be visualized by examining the PDF and CDF, which show that 86.5% of the area under the curve is less than x = 0.5.

Exponential PDF and CDF showing p(X < 0.5) for λ = 4

Exponential distribution in survival analysis[edit]

In survival analysis, the exponential distribution can be used to model the distribution of survival times for subjects in the study. The survivor function is the probability that a subject survives longer than time x. Because the CDF is the probability that a subject survives less than time x, the survivor function = 1 − CDF.

P(time between events is > 100) = 1 − P(time between events is < 100) = 1 − (1 − eλx) = eλx

The CDF graph below for the air conditioner failure example shows that P(x < 100) = 0.81. The survivor function graph shows that P(x > 100) = 1 − P(x < 100) = 1 – 0.81 = 0.19.

Survival function = 1 - CDF

Examples that violate the assumptions of an exponential distribution[edit]

The time between arrivals of students at the student union will likely not follow an exponential distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).

The time between magnitude 5 earthquakes per year in California may not follow an exponential distribution, if one large earthquake increases the probability of aftershocks of similar magnitude.

While the exponential distribution is sometimes appropriate to model survival, the actual distribution of lifespans in many settings do not follow the exponential distribution. In these situations, the Weibull or gamma distribution is commonly used, particularly for machines or devices. In medical research, survival is most commonly modeled using non-parametric or semi-parametric methods such as the Kaplan-Meier plot and Cox proportional hazards regression, rather than with parametric distributions such as the exponential or Weibull.

The graph shows a distribution of event times that is not exponential. The data are the age at first marriage of 5,534 US women who responded to the National Survey of Family Growth (NSFG) conducted by the CDC in the 2006 and 2010 cycle, from the data set ageAtMar in the R package openintro. In the example, the event is first marriage, and the time to event is age.

PDF and survival function for age at first marriage

These data violate the requirement that the rate at which events occur is constant. In these data, the rate is much higher in some intervals and lower in other intervals. The red line on the histogram shows the exponential curve fitted with lambda = 1/mean age. The survivor function graph is on the right and includes the 95% confidence interval in red. The survivor function and fitted exponential curve show that the decline from the initial value is not exponential.

The figure shows a Weibull distribution fit to the age at first marriage. While not a perfect fit, it is superior to the exponential fit, and would be more appropriate for modelling age at first marriage.

Weibull fit to distribution of age at first marriage

See also[edit]


  1. ^ David Olive, Chapter 4. Truncated Distributions, "Lemma 4.3", Southern Illinois University, February 18, 2010, p.107.
  2. ^ Template:Ref-missing
  4. ^ Michael, Lugo. "The expectation of the maximum of exponentials" (PDF). Retrieved 13 December 2016. 
  6. ^ NIST/SEMATECH e-Handbook of Statistical Methods
  7. ^ Abdulaziz Elfessi and David M. Reineke, "A Bayesian Look at Classical Estimation: The Exponential Distribution", Journal of Statistics Education Volume 9, Number 1 (2001).
  9. ^ Guerriero, V. (2012). "Power Law Distribution: Method of Multi-scale Inferential Statistics". Journal of Modern Mathematics Frontier (JMMF). 1: 21–28. 
  12. ^ "Cumfreq, a free computer program for cumulative frequency analysis". 
  14. ^ Lawless, J.F., Fredette, M.,"Frequentist predictions intervals and predictive distributions", Biometrika (2005), Vol 92, Issue 3, pp 529–542.
  17. ^ https://www.statlect.com/probability-distributions/exponential-distribution
  18. ^ Olkin, Ingram; Gleser, Leon; Derman, Cyrus (1994), Probability Models and Applications (Second ed.), Macmillan, ISBN 0-02-389220-X 

External links[edit]