In probability theory and statistics, the exponential distribution is the probability
distribution that describes the time between events in a Poisson process, i.e. a process
in which events occur continuously and independently at a constant average rate. 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 processes, it is found in various
other contexts. Note that the exponential distribution is
not the same as the class of exponential families of distributions, which is a large class of
probability distributions that includes the exponential distribution as one of its members,
but also includes the normal distribution, binomial distribution, gamma distribution,
Poisson, and many others.
Characterization Probability density function
The probability density function of an exponential distribution is
Alternatively, this can be defined using the Heaviside step function, H(x).
Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution
is supported on the interval [0, ∞). If a random variable X has this distribution,
we write X ~ Exp(λ). The exponential distribution exhibits infinite
divisibility. Differential equation
Cumulative distribution function The cumulative distribution function is given
by
Alternatively, this can be defined using the Heaviside step function, H(x).
Alternative parameterization A commonly used alternative parameterization
is to define the probability density function of an exponential distribution as
where β > 0 is a scale parameter of the distribution and is 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 parameterisation involving the
"rate" parameter arises in the context of events arriving at a rate λ, when the time
between events 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
or the notation in this section could be intended. 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 above, this makes sense: if you receive phone calls at
an average rate of 2 per hour, then you can expect to wait half an hour for every call.
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
where ln refers to the natural logarithm. Thus the absolute difference between the mean
and median is
in accordance with the median-mean inequality. Memorylessness
An exponentially distributed random variable T obeys the relation
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 unconditioned probability of observing the event more than 10 seconds
relative to 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 for Exp(λ) is
The quartiles are therefore: first quartile: ln(4λ
median: ln(2)/λ third quartile: ln(4)/λ
And as a consequence the interquartile range is ln(3)/λ.
Kullback–Leibler divergence The directed Kullback–Leibler divergence
of from is given by
Maximum entropy distribution Among all continuous probability distributions
with support and mean , the exponential distribution with has the largest differential entropy.
In other words, it is the maximum entropy probability distribution for a random variate
for which is fixed and greater than zero. Distribution of the 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 law
Note that
is not exponentially distributed. Parameter estimation
Suppose a given variable is exponentially distributed and the rate parameter λ is to
be estimated. Maximum likelihood
The likelihood function for λ, given an independent and identically distributed sample x = 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:
Although this is not an unbiased estimator of , is an unbiased estimator of where is
the scale parameter defined in the 'Alternative parameterization' section above.
Confidence intervals The 100(1 − α)% confidence interval for
the rate parameter of an exponential distribution is given by:
which is also equal to:
where χ2p,v is the 100(p) percentile of the chi squared distribution with v degrees of
freedom, n is the number of observations of inter-arrival times in the sample, and x-bar
is the sample average. A simple approximation to the exact interval endpoints can be derived
using a normal approximation to the χ2p,v distribution. This approximation gives the
following values for a 95% confidence interval:
This approximation may be acceptable for samples containing at least 15 to 20 elements.
Bayesian inference The conjugate prior for the exponential distribution
is the gamma distribution. The following parameterization of the gamma probability density function
is useful:
The posterior distribution p can then be expressed in terms of the likelihood function defined
above and a gamma prior:
Now the posterior density p has been specified up to a missing normalizing constant. Since
it has the form of a gamma pdf, this can easily be filled in, and one obtains:
Here the parameter α can be interpreted as the number of prior observations, and β as
the sum of the prior observations. The posterior mean here is:
Generating exponential variates A conceptually very simple method for generating
exponential variates is based on inverse transform sampling: Given a random variate U drawn from
the uniform distribution on the unit interval, the variate
has an exponential distribution, where F −1 is the quantile function, defined by
Moreover, if U is uniform on, then so is 1 − U. This means one can generate exponential
variates as follows:
Other methods for generating exponential variates are discussed by Knuth and Devroye.
The ziggurat algorithm is a fast method for generating exponential variates.
A fast method for generating a set of ready-ordered exponential variates without using a sorting
routine is also available. Related distributions
Exponential distribution is closed under scaling by a positive factor. If X ~ Exp(λ) then
kX ~ Exp(λ/k). If X ~ Exp(λ) and Y ~ Exp(ν) then min(X,
Y) ~ Exp(λ + ν). If Xi ~ Exp(λ) then min{X1, ..., Xn} ~ Exp(nλ).
The Benktander Weibull distribution reduces to a truncated exponential distribution. If
X ~ Exp(λ) then 1+X ~ BenktanderWeibull(λ, 1).
The exponential distribution is a limit of a scaled beta distribution:
If Xi ~ Exp(λ) then X1 + ... + Xk ~ Erlang(k, λ)
If X ~ Exp(1) then μ − σ log(X) ~ GEV(μ, σ, 0).
If X ~ Exp(λ) then X ~ Gamma(1, λ) If X ~ Exp(λ) and Y ~ Exp(ν) then λX − νY
~ Laplace(0, 1). If X, Y ~ Exp(λ) then X − Y ~ Laplace(0,
λ−1). If X ~ Laplace(μ, β−1) then |X − μ|
~ Exp(β). If X ~ Exp(1) then:
If X, Y ~ Exp(1) then:
If X ~ Exp(λ) then keX ~ Pareto(k, λ). If X ~ Pareto(1, λ) then log(X) ~ Exp(λ).
Exponential distribution is a special case of type 3 Pearson distribution.
If X ~ Exp(λ) then If X ~ Rayleigh(λ−1/2) then X2/2 ~ Exp(λ).
If X ~ Exp(λ) then If X ~ Exp(1) then
If Xi ~ U(0, 1) then
If Y|X ~ Poisson(X) where X ~ Exp(λ−1) then
If X ~ Exp(1) and then 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 Y ∼ Gumbel(μ, β). If X ~ Exp(1/2) then X ∼ χ22, i.e. X has
a chi-squared distribution with 2 degrees of freedom.
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 .
Other 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.
Applications Occurrence of events
The exponential distribution occurs naturally when describing the lengths of the inter-arrival
times in a homogeneous Poisson process. The exponential distribution may be viewed
as a continuous counterpart of the geometric distribution, which describes the number of
Bernoulli trials necessary for a discrete process to change state. In contrast, the
exponential distribution describes the time for a continuous process to change state.
In real-world scenarios, the assumption of a constant rate is rarely satisfied. For example,
the rate of incoming phone calls differs according to the time of day. But if we focus on a time
interval during which the rate is roughly constant, such as from 2 to 4 p.m. during
work days, the exponential distribution can be used as a good approximate model for the
time until the next phone call arrives. Similar caveats apply to the following examples which
yield approximately exponentially distributed variables:
The time until a radioactive particle decays, or the time between clicks of a geiger counter
The time it takes before your next telephone call
The time until default in reduced form credit risk modeling
Exponential variables can also be used to model situations where certain events occur
with a constant probability per unit length, such as the distance between mutations on
a DNA strand, or between roadkills on a given road.
In queuing theory, the service times of agents in a system are often modeled as exponentially
distributed variables. The length of a process that can be thought of as a sequence of several
independent tasks follows the Erlang distribution.
Reliability theory and reliability engineering also make extensive use of the exponential
distribution. Because of the memoryless property of this distribution, it is well-suited to
model the constant hazard rate portion of the bathtub curve used in reliability theory.
It is also very convenient because it is so easy to add failure rates in a reliability
model. The exponential distribution is however not appropriate to model the overall lifetime
of organisms or technical devices, because the "failure rates" here are not constant:
more failures occur for very young and for very old systems.
In physics, if you observe a gas at a fixed temperature and pressure in a uniform gravitational
field, the heights of the various molecules also follow an approximate exponential distribution,
known as the Barometric formula. This is a consequence of the entropy property mentioned
below. In hydrology, the exponential distribution
is used to analyze extreme values of such variables as monthly and annual maximum values
of daily rainfall and river discharge volumes. The blue picture illustrates an example of
fitting the exponential distribution to ranked annually maximum one-day rainfalls showing
also the 90% confidence belt based on the binomial distribution. The rainfall data are
represented by plotting positions as part of the cumulative frequency analysis.
Prediction Having observed a sample of n data points
from an unknown exponential distribution a common task is to use these samples to make
predictions about future data from the same source. A common predictive distribution over
future samples is the so-called plug-in distribution, formed by plugging a suitable estimate for
the rate parameter λ into the exponential density function. A common choice of estimate
is the one provided by the principle of maximum likelihood, and using this yields the predictive
density over a future sample xn+1, conditioned on the observed samples x = given by
The Bayesian approach provides a predictive distribution which takes into account the
uncertainty of the estimated parameter, although this may depend crucially on the choice of
prior. A predictive distribution free of the issues
of choosing priors that arise under the subjective Bayesian approach is
which can be considered as (1) a frequentist confidence distribution,
obtained from the distribution of the pivotal quantity ;
(2) a profile predictive likelihood, obtained by eliminating the parameter λ from the joint
likelihood of xn+1 and λ by maximization; (3) an objective Bayesian predictive posterior
distribution, obtained using the non-informative Jeffreys prior 1/λ;
(4) the Conditional Normalized Maximum Likelihood predictive distribution, from information
theoretic considerations. The accuracy of a predictive distribution
may be measured using the distance or divergence between the true exponential distribution
with rate parameter, λ0, and the predictive distribution based on the sample x. The Kullback–Leibler
divergence is a commonly used, parameterisation free measure of the difference between two
distributions. Letting Δ(λ0||p) denote the Kullback–Leibler divergence between an exponential
with rate parameter λ0 and a predictive distribution p it can be shown that
where the expectation is taken with respect to the exponential distribution with rate
parameter λ0 ∈, and ψ( · ) is the digamma function. It is clear that the CNML predictive
distribution is strictly superior to the maximum likelihood plug-in distribution in terms of
average Kullback–Leibler divergence for all sample sizes n > 0.
See also Dead time – an application of exponential
distribution to particle detector analysis. Laplace distribution, or the "double exponential
distribution". References
External links Hazewinkel, Michiel, ed., "Exponential distribution",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Online calculator of Exponential Distribution