exponential distribution
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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the exponential distribution or negative exponential distribution is the
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
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 In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
, and it has the key property of being
memoryless In probability and statistics, memorylessness is a property of probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential d ...
. 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, like the normal, binomial,
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
, and Poisson distributions.


Definitions


Probability density function

The
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
(pdf) of an exponential distribution is : f(x;\lambda) = \begin \lambda e^ & x \ge 0, \\ 0 & x < 0. \end 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
''X'' has this distribution, we write . The exponential distribution exhibits infinite divisibility.


Cumulative distribution function

The
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
is given by :F(x;\lambda) = \begin 1-e^ & x \ge 0, \\ 0 & x < 0. \end


Alternative parametrization

The exponential distribution is sometimes parametrized in terms of the scale parameter , which is also the mean: f(x;\beta) = \begin \frac e^ & x \ge 0, \\ 0 & x < 0. \end \qquad\qquad F(x;\beta) = \begin 1- e^ & x \ge 0, \\ 0 & x < 0. \end


Properties


Mean, variance, moments, and median

The mean or
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of an exponentially distributed random variable ''X'' with rate parameter ''λ'' is given by \operatorname = \frac. 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 In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
of ''X'' is given by \operatorname = \frac, so the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
is equal to the mean. The moments of ''X'', for n\in\N are given by \operatorname\left ^n\right= \frac. The central moments of ''X'', for n\in\N are given by \mu_n = \frac = \frac\sum^n_\frac. where !''n'' is the subfactorial of ''n'' The
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
of ''X'' is given by \operatorname = \frac < \operatorname where refers to the
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
. Thus the absolute difference between the mean and median is \left, \operatorname\left \right- \operatorname\left \right = \frac < \frac = \operatorname in accordance with the median-mean inequality.


Memorylessness property of exponential random variable

An exponentially distributed random variable ''T'' obeys the relation \Pr \left (T > s + t \mid T > s \right ) = \Pr(T > t), \qquad \forall s, t \ge 0. This can be seen by considering the complementary cumulative distribution function: \begin \Pr\left(T > s + t \mid T > s\right) &= \frac \\ pt &= \frac \\ pt &= \frac \\ pt &= e^ \\ pt &= \Pr(T > t). \end 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 In probability theory, conditional probability is a measure of the probability of an Event (probability theory), event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This ...
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 In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
are the only memoryless probability distributions. The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant
failure rate Failure is the social concept of not meeting a desirable or intended objective, and is usually viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. On ...
.


Quantiles

The quantile function (inverse cumulative distribution function) for Exp(''λ'') is F^(p;\lambda) = \frac,\qquad 0 \le p < 1 The quartiles are therefore: * first quartile: ln(4/3)/''λ'' *
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
: ln(2)/''λ'' * third quartile: ln(4)/''λ'' And as a consequence the
interquartile range In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the differen ...
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: \begin \bar_\alpha (X) &= \frac \int_^ q_p (X) dp \\ &= \frac \int_^ \frac dp \\ &= \frac \int_^ -\ln (y ) dy \\ &= \frac \int_^ \ln (y ) dy \\ &= \frac ( 1-\alpha) \ln(1-\alpha) - (1-\alpha) \\ &= \frac \\ \end


Buffered Probability of Exceedance (bPOE)

The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold x. It is derived as follows: \begin \bar_x (X) &= \ \\ &= \ \\ &= \ \\ &= \ = \ = e^ \end


Kullback–Leibler divergence

The directed
Kullback–Leibler divergence In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how much a model probability distribution is diff ...
in nats of e^\lambda ("approximating" distribution) from e^ ('true' distribution) is given by \begin \Delta(\lambda_0 \parallel \lambda) &= \mathbb_\left( \log \frac\right)\\ &= \mathbb_\left( \log \frac\right)\\ &= \log(\lambda_0) - \log(\lambda) - (\lambda_0 - \lambda)E_(x)\\ &= \log(\lambda_0) - \log(\lambda) + \frac - 1. \end


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.


Distribution of the minimum of exponential random variables

Let ''X''1, ..., ''X''''n'' be independent exponentially distributed random variables with rate parameters ''λ''1, ..., ''λn''. Then \min\left\ is also exponentially distributed, with parameter \lambda = \lambda_1 + \dotsb + \lambda_n. This can be seen by considering the complementary cumulative distribution function: \begin &\Pr\left(\min\ > x\right) \\ = &\Pr\left(X_1 > x, \dotsc, X_n > x\right) \\ = &\prod_^n \Pr\left(X_i > x\right) \\ = &\prod_^n \exp\left(-x\lambda_i\right) = \exp\left(-x\sum_^n \lambda_i\right). \end The index of the variable which achieves the minimum is distributed according to the categorical distribution \Pr\left(X_k = \min\\right) = \frac. A proof can be seen by letting I = \operatorname_\. Then, \begin \Pr (I = k) &= \int_^ \Pr(X_k = x) \Pr(\forall_X_ > x ) \,dx \\ &= \int_^ \lambda_k e^ \left(\prod_^ e^\right) dx \\ &= \lambda_k \int_^ e^ dx \\ &= \frac. \end Note that \max\ is not exponentially distributed, if ''X''1, ..., ''X''''n'' do not all have parameter 0.


Joint moments of i.i.d. exponential order statistics

Let X_1, \dotsc, X_n be n independent and identically distributed exponential random variables with rate parameter ''λ''. Let X_, \dotsc, X_ denote the corresponding
order statistic In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with Ranking (statistics), rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and ...
s. For i < j , the joint moment \operatorname E\left _ X_\right of the order statistics X_ and X_ is given by \begin \operatorname E\left _ X_\right &= \sum_^\frac \operatorname E\left _\right+ \operatorname E\left _^2\right\\ &= \sum_^\frac\sum_^\frac + \sum_^\frac + \left(\sum_^\frac\right)^2. \end This can be seen by invoking the
law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property of conditional expectation, among other names, states that if X is a random ...
and the memoryless property: \begin \operatorname E\left _ X_\right &= \int_0^\infty \operatorname E\left _ X_ \mid X_=x\rightf_(x) \, dx \\ &= \int_^\infty x \operatorname E\left _ \mid X_ \geq x\rightf_(x) \, dx &&\left(\textrm~X_ = x \implies X_ \geq x\right) \\ &= \int_^\infty x \left _\right+ x \right">\operatorname E\left _\right+ x \rightf_(x) \, dx &&\left(\text\right) \\ &= \sum_^\frac \operatorname E\left _\right+ \operatorname E\left _^2\right \end The first equation follows from the
law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing property of conditional expectation, among other names, states that if X is a random ...
. The second equation exploits the fact that once we condition on X_ = x , it must follow that X_ \geq x . The third equation relies on the memoryless property to replace \operatorname E\left X_ \mid X_ \geq x\right/math> with \operatorname E\left _\right+ x.


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 X_1 and X_2 are independent exponential random variables with respective rate parameters \lambda_1 and \lambda_2, then the probability density of Z=X_1+X_2 is given by \begin f_Z(z) &= \int_^\infty f_(x_1) f_(z - x_1)\,dx_1\\ &= \int_0^z \lambda_1 e^ \lambda_2 e^ \, dx_1 \\ &= \lambda_1 \lambda_2 e^ \int_0^z e^\,dx_1 \\ &= \begin \dfrac \left(e^ - e^\right) & \text \lambda_1 \neq \lambda_2 \\ pt \lambda^2 z e^ & \text \lambda_1 = \lambda_2 = \lambda. \end \end The entropy of this distribution is available in closed form: assuming \lambda_1 > \lambda_2 (without loss of generality), then \begin H(Z) &= 1 + \gamma + \ln \left( \frac \right) + \psi \left( \frac \right) , \end where \gamma is the Euler-Mascheroni constant, and \psi(\cdot) is the
digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...
. In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter \lambda, 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

* If ''X'' ~ Laplace(μ, β−1), then , ''X'' − μ, ~ Exp(β). * If ''X'' ~ ''U''(0, 1) then −log(''X'') ~ Exp(1). * If ''X'' ~ Pareto(1, λ), then log(''X'') ~ Exp(λ). * If ''X'' ~ SkewLogistic(θ), then \log\left(1 + e^\right) \sim \operatorname(\theta). * If ''Xi'' ~ ''U''(0, 1) then \lim_n \min \left(X_1, \ldots, X_n\right) \sim \operatorname(1) * The exponential distribution is a limit of a scaled
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...
: \lim_ n \operatorname(1, n) = \operatorname(1). * The exponential distribution is a special case of type 3 Pearson distribution. * The exponential distribution is the special case of a Gamma distribution with shape parameter 1. * If ''X'' ~ Exp(λ) and ''X'' ~ Exp(λ) then: ** kX \sim \operatorname\left(\frac\right), closure under scaling by a positive factor. ** 1 + ''X'' ~ BenktanderWeibull(λ, 1), which reduces to a truncated exponential distribution. ** ''keX'' ~ Pareto(''k'', λ). ** ''e−λX'' ~ ''U''(0, 1). ** ''e−X'' ~
Beta Beta (, ; uppercase , lowercase , or cursive ; or ) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represe ...
(λ, 1). ** ''e'' ~ PowerLaw(''k'', λ) ** \sqrt \sim \operatorname \left(\frac\right), the Rayleigh distribution ** X \sim \operatorname\left(\frac, 1\right), the Weibull distribution ** X^2 \sim \operatorname\left(\frac, \frac\right) ** . ** \lfloor X\rfloor \sim \operatorname\left(1-e^\right), a
geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
on 0,1,2,3,... ** \lceil X\rceil \sim \operatorname\left(1-e^\right), a
geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
on 1,2,3,4,... ** If also ''Y'' ~ Erlang(''n'', λ) orY \sim \Gamma\left(n, \frac\right) then \frac + 1 \sim \operatorname(1, n) ** If also λ ~
Gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
(''k'', θ) (shape, scale parametrisation) then the marginal distribution of ''X'' is Lomax(''k'', 1/θ), the gamma
mixture In chemistry, a mixture is a material made up of two or more different chemical substances which can be separated by physical method. It is an impure substance made up of 2 or more elements or compounds mechanically mixed together in any proporti ...
** λ''X'' − λ''Y'' ~ Laplace(0, 1). ** min ~ Exp(λ1 + ... + λ''n''). ** If also λ = λ then: *** X_1 + \cdots + X_k = \sum_i X_i \sim Erlang(''k'', λ) =
Gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
(''k'', λ) with integer shape parameter ''k'' and rate parameter λ. *** If T = (X_1 + \cdots + X_n ) = \sum_^n X_i, then 2 \lambda T \sim \chi^2_. *** ''X'' − ''X'' ~ Laplace(0, λ−1). ** If also ''X'' are independent, then: *** \frac ~ U(0, 1) *** Z = \frac has probability density function f_Z(z) = \frac. This can be used to obtain a confidence interval for \frac. ** If also λ = 1: *** \mu - \beta\log\left(\frac\right) \sim \operatorname(\mu, \beta), the logistic distribution *** \mu - \beta\log\left(\frac\right) \sim \operatorname(\mu, \beta) *** ''μ'' − σ log(''X'') ~ GEV(μ, σ, 0). *** Further if Y \sim \Gamma\left(\alpha, \frac\right) then \sqrt \sim \operatorname(\alpha, \beta) ( K-distribution) ** If also λ = 1/2 then ; i.e., ''X'' has a
chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...
with 2
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
. Hence: \operatorname(\lambda) = \frac \operatorname\left(\frac \right) \sim \frac \chi_2^2\Rightarrow \sum_^n \operatorname(\lambda) \sim \frac\chi_^2 * If X \sim \operatorname\left(\frac\right) and Y \mid X ~ Poisson(''X'') then Y \sim \operatorname\left(\frac\right) (
geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
) * 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 \kappa = 0 case. * Exponential distribution is a limit of the κ-Generalized Gamma distribution in the \alpha = 1 and \nu = 1 cases: *: \lim_ p_\kappa(x) = (1+\kappa\nu)(2\kappa)^\nu \frac \frac x^\exp_\kappa(-\lambda x^\alpha) = \lambda e^ 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 In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
.


Statistical inference

Below, suppose random variable ''X'' is exponentially distributed with rate parameter λ, and x_1, \dotsc, x_n are ''n'' independent samples from ''X'', with sample mean \bar.


Parameter estimation

The maximum likelihood estimator for λ is constructed as follows. The
likelihood function A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...
for λ, given an independent and identically distributed sample ''x'' = (''x''1, ..., ''x''''n'') drawn from the variable, is: L(\lambda) = \prod_^n\lambda\exp(-\lambda x_i) = \lambda^n\exp\left(-\lambda \sum_^n x_i\right) = \lambda^n\exp\left(-\lambda n\overline\right), where: \overline = \frac\sum_^n x_i is the sample mean. The derivative of the likelihood function's logarithm is: \frac \ln L(\lambda) = \frac \left( n \ln\lambda - \lambda n\overline \right) = \frac - n\overline\ \begin > 0, & 0 < \lambda < \frac, \\ pt = 0, & \lambda = \frac, \\ pt < 0, & \lambda > \frac. \end Consequently, the maximum likelihood estimate for the rate parameter is: \widehat_\text = \frac = \frac This is an
unbiased estimator In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In stat ...
of \lambda, although \overline an unbiased MLE estimator of 1/\lambda and the distribution mean. The bias of \widehat_\text is equal to B \equiv \operatorname\left left(\widehat_\text - \lambda\right)\right= \frac which yields the bias-corrected maximum likelihood estimator \widehat^*_\text = \widehat_\text - B. An approximate minimizer of
mean squared error In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference betwee ...
(see also:
bias–variance tradeoff In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions, and how well it can make predictions on previously unseen data that were not used to train ...
) can be found, assuming a sample size greater than two, with a correction factor to the MLE: \widehat = \left(\frac\right) \left(\frac\right) = \frac This is derived from the mean and variance of the inverse-gamma distribution, \mbox(n, \lambda).


Fisher information

The Fisher information, denoted \mathcal(\lambda), for an estimator of the rate parameter \lambda is given as: \mathcal(\lambda) = \operatorname \left \lambda\right= \int \left(\frac \log f(x;\lambda)\right)^2 f(x; \lambda)\,dx Plugging in the distribution and solving gives: \mathcal(\lambda) = \int_^ \left(\frac \log \lambda e^\right)^2 \lambda e^\,dx = \int_^ \left(\frac - x\right)^2 \lambda e^\,dx = \lambda^. This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter \lambda.


Confidence intervals

An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by: \frac< \frac < \frac\,, which is also equal to \frac < \frac < \frac\,, where is the
percentile In statistics, a ''k''-th percentile, also known as percentile score or centile, is a score (e.g., a data point) a given percentage ''k'' of all scores in its frequency distribution exists ("exclusive" definition) or a score a given percentage ...
of the chi squared distribution with ''v''
degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
, n is the number of observations and x-bar is the sample average. A simple approximation to the exact interval endpoints can be derived using a normal approximation to the distribution. This approximation gives the following values for a 95% confidence interval: \begin \lambda_\text &= \widehat\left(1 - \frac\right) \\ \lambda_\text &= \widehat\left(1 + \frac\right) \end 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 (of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful: \operatorname(\lambda; \alpha, \beta) = \frac \lambda^ \exp(-\lambda\beta). The posterior distribution ''p'' can then be expressed in terms of the likelihood function defined above and a gamma prior: \begin p(\lambda) &\propto L(\lambda) \Gamma(\lambda; \alpha, \beta) \\ &= \lambda^n \exp\left(-\lambda n\overline\right) \frac \lambda^ \exp(-\lambda \beta) \\ &\propto \lambda^ \exp(-\lambda \left(\beta + n\overline\right)). \end 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: p(\lambda) = \operatorname(\lambda; \alpha + n, \beta + n\overline). Here the hyperparameter ''α'' can be interpreted as the number of prior observations, and ''β'' as the sum of the prior observations. The posterior mean here is: \frac.


Occurrence and applications


Occurrence of events

The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous
Poisson process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points ...
. The exponential distribution may be viewed as a continuous counterpart of the
geometric distribution In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: * The probability distribution of the number X of Bernoulli trials needed to get one success, supported on \mathbb = \; * T ...
, which describes the number of
Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...
s 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 (or probability per unit time) 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 decay In particle physics, particle decay is the spontaneous process of one unstable subatomic particle transforming into multiple other particles. The particles created in this process (the ''final state'') must each be less massive than the original ...
s, or the time between clicks of a Geiger counter * The time between receiving one telephone call and the next * The time until default (on payment to company debt holders) 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
mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, ...
s on a
DNA Deoxyribonucleic acid (; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of al ...
strand, or between roadkills on a given road. In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. (The arrival of customers for instance is also modeled by the
Poisson distribution In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known const ...
if the arrivals are independent and distributed identically.) The length of a process that can be thought of as a sequence of several independent tasks follows the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).
Reliability theory Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability is defined as the probability that a product, system, or service will perform its intended funct ...
and
reliability engineering Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability is defined as the probability that a product, system, or service will perform its intended functi ...
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 rate Failure is the social concept of not meeting a desirable or intended objective, and is usually viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. On ...
s 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 Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, if you observe a gas at a fixed
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
in a uniform
gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...
, 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 Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
, 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 In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis. In operating-rooms management, the distribution of surgery duration for a category of surgeries with no typical work-content (like in an emergency room, encompassing all types of surgeries).


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 ''x''''n''+1, conditioned on the observed samples ''x'' = (''x''1, ..., ''xn'') given by p_(x_ \mid x_1, \ldots, x_n) = \left( \frac1 \right) \exp \left( - \frac \right). 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 p_(x_ \mid x_1, \ldots, x_n) = \frac, which can be considered as # a frequentist confidence distribution, obtained from the distribution of the pivotal quantity /; # a profile predictive likelihood, obtained by eliminating the parameter ''λ'' from the joint likelihood of ''x''''n''+1 and ''λ'' by maximization; # an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/''λ''; # the Conditional Normalized Maximum Likelihood (CNML) 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, ''λ''0, and the predictive distribution based on the sample ''x''. The
Kullback–Leibler divergence In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how much a model probability distribution is diff ...
is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(''λ''0, , ''p'') denote the Kullback–Leibler divergence between an exponential with rate parameter ''λ''0 and a predictive distribution ''p'' it can be shown that \begin \operatorname_ \left \Delta(\lambda_0\parallel p_) \right&= \psi(n) + \frac - \log(n) \\ \operatorname_ \left \Delta(\lambda_0\parallel p_) \right&= \psi(n) + \frac - \log(n) \end where the expectation is taken with respect to the exponential distribution with rate parameter , and 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 .


Random variate generation

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 T = F^(U) has an exponential distribution, where ''F'' is the quantile function, defined by F^(p)=\frac. Moreover, if ''U'' is uniform on (0, 1), then so is 1 − ''U''. This means one can generate exponential variates as follows: T = \frac. Other methods for generating exponential variates are discussed by Knuth Donald E. Knuth (1998). '' The Art of Computer Programming'', volume 2: ''Seminumerical Algorithms'', 3rd edn. Boston: Addison–Wesley. . ''See section 3.4.1, p. 133.'' and Devroye.Luc Devroye (1986).
Non-Uniform Random Variate Generation
'. New York: Springer-Verlag. . ''Se
chapter IX
section 2, pp. 392–401.''
A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.


See also

* Dead time – an application of exponential distribution to particle detector analysis. *
Laplace distribution In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
, or the "double exponential distribution". * Relationships among probability distributions * Marshall–Olkin exponential distribution


References


External links

*
Online calculator of Exponential Distribution
{{DEFAULTSORT:Exponential Distribution Continuous distributions Exponentials Poisson point processes Conjugate prior distributions Exponential family distributions Infinitely divisible probability distributions Survival analysis