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, such as the normal, binomial, gamma, and Poisson distributions.
Definitions
Probability density function The probability density function (pdf) of an exponential distribution is
Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval . If a random variable X has this distribution, we write .
The exponential distribution exhibits infinite divisibility.
Cumulative distribution function The cumulative distribution function is given by
Alternative parametrization The exponential distribution is sometimes parametrized in terms of the scale parameter , which is also the mean:
Properties
Mean, variance, moments, and median
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
In light of the examples given below, this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes.
The variance of X is given by
so the standard deviation is equal to the mean.
The moments of X, for are given by
The central moments of X, for are given by
where !n is the subfactorial of n.
The median of X is given by
where refers to the natural logarithm. Thus the absolute difference between the mean and median is
in accordance with the median-mean inequality.
Memorylessness property of exponential random variable An exponentially distributed random variable T obeys the relation
This can be seen by considering the complementary cumulative distribution function:
When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the conditional probability that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds after the initial time.
The exponential distribution and the geometric distribution are the only memoryless probability distributions.
The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate.
Quantiles
The quantile function (inverse cumulative distribution function) for Exp(λ) is
The quartiles are therefore: first quartile: ln(4/3)/λ median: ln(2)/λ third quartile: ln(4)/λ
And as a consequence the interquartile range is ln(3)/λ.
Conditional Value at Risk (Expected Shortfall) The conditional value at risk (CVaR) also known as the expected shortfall or superquantile for Exp(λ) is derived as follows:
Buffered Probability of Exceedance (bPOE)
The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold . It is derived as follows:
Kullback–Leibler divergence The directed Kullback–Leibler divergence in nats of ("approximating" distribution) from ('true' distribution) is given by
Maximum entropy distribution Among all continuous probability distributions with support and mean μ, the exponential distribution with λ = 1/μ has the largest differential entropy. In other words, it is the maximum entropy probability distribution for a random variate X which is greater than or equal to zero and for which E[X] is fixed.
Multiplication with a constant
For a random variable with parameter , the variable multiplied with a constant is again exponentially distributed:
Minimum of exponential random variables Let X1, ..., Xn be independent exponentially distributed random variables with rate parameters λ1, ..., λn. Then
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 categorical distribution
A proof can be seen by letting . Then,
In contrast, is not exponentially distributed, if X1, ..., Xn do not all have parameter 0.
Joint moments of i.i.d. exponential order statistics
Let be independent and identically distributed exponential random variables with rate parameter λ. Let denote the corresponding order statistics. For , the joint moment of the order statistics and is given by
This can be seen by invoking the law of total expectation and the memoryless property:
The first equation follows from the law of total expectation. The second equation exploits the fact that once we condition on , it must follow that . The third equation relies on the memoryless property to replace with .
Sum of two independent exponential random variables The probability distribution function (PDF) of a sum of two independent random variables is the convolution of their individual PDFs. If and are independent exponential random variables with respective rate parameters and then the probability density of is given by
The entropy of this distribution is available in closed form: assuming (without loss of generality), then
where is the Euler-Mascheroni constant, and is the digamma function.
In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter which in turn is a special case of gamma distribution.
The sum of n independent Exp(λ) exponential random variables is Gamma(n, λ) distributed.
Related distributions
In this section, the following conventions are used:
for the (continuous) uniform distribution on the interval with . for the beta distribution with parameters . for the Poisson distribution with parameter . for the Gamma distribution with shape parameter and rate parameter . for the Erlang distribution with shape parameter and rate parameter . for the chi-squared distribution with degrees of freedom. the Pareto distribution with scale parameter and shape parameter . for the Laplace distribution with location and scale parameter . for the type I generalized logistic distribution with parameter . for the Reyleigh distribution with parameter . for the Weibull distribution with scale and shape parameters . for the Gumbel distribution with location parameter and scale parameter . for the logistic distribution with location parameter and scale parameter . the geometric distribution with success probability and support . the geometric distribution with success probability and support . for the Lomax distribution with shape parameter and scale parameter . for the generalized extreme value distribution with location parameter , scale parameter and shape parameter . for the Benktander type II distribution with parameters and .
Special cases of other distributions
The exponential distribution is a special case of type 3 Pearson distribution. For any : . .
Functional relationships to other distributions If X ~ Laplace(μ, β), then |X − μ| ~ Exp(β). On the other hand, if X, Y ~ Exp(λ) are independent, then X - Y ~ Laplace(0, λ). Also, if X ~ Exp(λ) and Y ~ Exp(λ) are independent, then λX - λY ~ Laplace(0, 1). If X ~ Beta(λ, 1), then −log(X) ~ Exp(λ). Equivalently, if Y ~ Exp(λ), then e−Y ~ Beta(λ, 1). As a special case of the above: If X ~ U(0, 1) then −log(X)/λ ~ Exp(λ). Equivalently, if Y ~ Exp(λ), then e−λY ~ U(0, 1). This is a direct consequence of the probability integral transform. If X ~ Pareto(k, λ), then log(X/k) ~ Exp(λ). Equivalently, if Y ~ Exp(λ), then keY ~ Pareto(k, λ). If X ~ SkewLogistic(θ), then . Equivalently, if Y ~ Exp(λ), then −log(e−1) ~ SkewLogistic(λ). If X ~ Rayleigh(σ), then . Equivalently, if Y ~ Exp(λ), then . If X ~ Weibull(λ, k), then X ~ Exp(1/λ). In particular, if Y ~ Exp(λ), then Y ~ Weibull(1/λ, 1/2). If X ~ Gumbel(μ, β), then exp((X-μ)/β) ~ Exp(1). Equivalently, if Y ~ Exp(λ), then = GEV(μ, σ, 0). If X ~ BenktanderWeibull(λ, 1), then X - 1 ~ Exp(λ). Equivalently, if Y ~ Exp(λ), then 1 + Y ~ BenktanderWeibull(λ, 1), which reduces to a truncated exponential distribution. If X ~ Exp(λ), then , and e ~ PowerLaw(k, λ). If X ~ Exp(1), then . If X, Y ~ Exp(1) are independent, then X/Y ~ Lomax(1, 1) and . The former can be used to obtain a confidence interval for quotients of rates . If X ~ Exp(1) and are independent, then . If X ~ Exp(λ) and λ ~ Gamma(k, Λ), then the marginal distribution of X is Lomax(k, Λ), the gamma mixture. If X ~ Exp(λ) and ~ Poisson(X), then . If Yi ~ U(0, 1), thenSince , the exponential distribution is therefore a (weak) limit of a scaled beta distribution: The Hoyt distribution can be obtained from exponential distribution and arcsine distribution. The exponential distribution is a limit of the κ-exponential distribution in the case. Exponential distribution is a limit of the κ-Generalized Gamma distribution in the and cases:
Indirectly related distributions: Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities. Hypoexponential distribution – the distribution of a general sum of exponential random variables. exGaussian distribution – the sum of an exponential distribution and a normal distribution.
Sequences of exponential variables
In the following, let X, X, ..., X ~ Exp(λ) be all independent.
Erlang(n, λ) = Gamma(n, λ). If , then . Lomax(n, 1) and Pareto(1, n). Beta(1, n). In particular, U(0,1). Statistical inference Below, suppose random variable X is exponentially distributed with rate parameter λ, and are n independent samples from X, with sample mean .
Parameter estimation
Maximum Likelihood estimator
The maximum likelihood estimator for λ is constructed as follows.
The likelihood function for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from the variable, is:
where:
is the sample mean.
The derivative of the likelihood function's logarithm is:
Consequently, the maximum likelihood estimate for the rate parameter is:
This is an unbiased estimator of although an unbiased MLENIST/SEMATECH e-Handbook of Statistical Methods of and the distribution mean. There are methods to reduce the bias of yielding a bias-corrected maximum likelihood estimator.
An approximate minimizer of the mean squared error (see also: bias–variance tradeoff) can be found, assuming a sample size greater than two, with a correction factor to the MLE:
This is derived from the mean and variance of the inverse-gamma distribution, .
UMVUE
The uniformly minimum-variance unbiased estimator for is
which, in contrast to the MLE, is unbiased by definition.
Fisher information The Fisher information, denoted , for an estimator of the rate parameter is given as:
Plugging in the distribution and solving gives:
This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter .