In

_{1}, â€¦, ''X''_{''n''} be Independent random variables, independent exponentially distributed random variables with rate parameters ''Î»''_{1}, â€¦, ''Î»_{n}''. Then
$$\backslash min\backslash left\backslash $$
is also exponentially distributed, with parameter
$$\backslash lambda\; =\; \backslash lambda\_1\; +\; \backslash dotsb\; +\; \backslash lambda\_n.$$
This can be seen by considering the Cumulative distribution function#Complementary cumulative distribution function (tail distribution), complementary cumulative distribution function:
$$\backslash begin\; \&\backslash Pr\backslash left(\backslash min\backslash \; >\; x\backslash right)\; \backslash \backslash \; =\; \&\backslash Pr\backslash left(X\_1\; >\; x,\; \backslash dotsc,\; X\_n\; >\; x\backslash right)\; \backslash \backslash \; =\; \&\backslash prod\_^n\; \backslash Pr\backslash left(X\_i\; >\; x\backslash right)\; \backslash \backslash \; =\; \&\backslash prod\_^n\; \backslash exp\backslash left(-x\backslash lambda\_i\backslash right)\; =\; \backslash exp\backslash left(-x\backslash sum\_^n\; \backslash lambda\_i\backslash right).\; \backslash end$$
The index of the variable which achieves the minimum is distributed according to the categorical distribution
$$\backslash Pr\backslash left(k\; \backslash mid\; X\_k\; =\; \backslash min\backslash \backslash right)\; =\; \backslash frac.$$
A proof can be seen by letting $I\; =\; \backslash operatorname\_\backslash $. Then,
$$\backslash begin\; \backslash Pr\; (I\; =\; k)\; \&=\; \backslash int\_^\; \backslash Pr(X\_k\; =\; x)\; \backslash Pr(X\_\; >\; x\; )\; \backslash ,dx\; \backslash \backslash \; \&=\; \backslash int\_^\; \backslash lambda\_k\; e^\; \backslash left(\backslash prod\_^\; e^\backslash right)\; dx\; \backslash \backslash \; \&=\; \backslash lambda\_k\; \backslash int\_^\; e^\; dx\; \backslash \backslash \; \&=\; \backslash frac.\; \backslash end$$
Note that
$$\backslash max\backslash $$
is not exponentially distributed, if ''X''_{1}, â€¦, ''X''_{''n''} do not all have parameter 0.

_{i}'' ~ Uniform distribution (continuous), ''U''(0, 1) then $$\backslash lim\_n\; \backslash min\; \backslash left(X\_1,\; \backslash ldots,\; X\_n\backslash right)\; \backslash sim\; \backslash operatorname(1)$$
* The exponential distribution is a limit of a scaled beta distribution: $$\backslash lim\_\; n\; \backslash operatorname(1,\; n)\; =\; \backslash operatorname(1).$$
* Exponential distribution is a special case of type 3 Pearson distribution.
* If ''X'' ~ Exp(Î») and ''X'' ~ Exp(Î») then:
** $kX\; \backslash sim\; \backslash operatorname\backslash left(\backslash frac\backslash right)$, closure under scaling by a positive factor.
** 1 + ''X'' ~ Benktander Weibull distribution, BenktanderWeibull(Î», 1), which reduces to a truncated exponential distribution.
** ''ke^{X}'' ~ Pareto distribution, Pareto(''k'', Î»).
** ''e^{âˆ’X}'' ~ Beta distribution, Beta(Î», 1).
** ''e'' ~ power law, PowerLaw(''k'', Î»)
** $\backslash sqrt\; \backslash sim\; \backslash operatorname\; \backslash left(\backslash frac\backslash right)$, the Rayleigh distribution
** $X\; \backslash sim\; \backslash operatorname\backslash left(\backslash frac,\; 1\backslash right)$, the Weibull distribution
** $X^2\; \backslash sim\; \backslash operatorname\backslash left(\backslash frac,\; \backslash frac\backslash right)$
** .
** $\backslash lfloor\; X\backslash rfloor\; \backslash sim\; \backslash operatorname\backslash left(1-e^\backslash right)$, a geometric distribution
In probability theory
Probability theory 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 expres ...

on 1,2,3,4,...
** If also ''Y'' ~ Erlang(''n'', Î») or$Y\; \backslash sim\; \backslash Gamma\backslash left(n,\; \backslash frac\backslash right)$ then $\backslash frac\; +\; 1\; \backslash sim\; \backslash operatorname(1,\; n)$
** If also Î» ~ gamma distribution, Gamma(''k'', Î¸) (shape, scale parametrisation) then the marginal distribution of ''X'' is Lomax distribution, Lomax(''k'', 1/Î¸), the gamma compound distribution, mixture
** Î»''X'' âˆ’ Î»''Y'' ~ Laplace distribution, Laplace(0, 1).
** min ~ Exp(Î»_{1} + ... + Î»_{''n''}).
** If also Î» = Î» then:
*** $X\_1\; +\; \backslash cdots\; +\; X\_k\; =\; \backslash sum\_i\; X\_i\; \backslash sim$ Erlang distribution, Erlang(''k'', Î») = gamma distribution, Gamma(''k'', Î»^{âˆ’1}) = Gamma(''k'', Î») (in (''k'', Î¸) and (Î±, Î²) parametrization, respectively) with an integer shape parameter k.
*** ''X'' âˆ’ ''X'' ~ Laplace(0, Î»^{âˆ’1}).
** If also ''X'' are independent, then:
*** $\backslash frac$ ~ uniform distribution (continuous), U(0, 1)
*** $Z\; =\; \backslash frac$ has probability density function $f\_Z(z)\; =\; \backslash frac$. This can be used to obtain a confidence interval for $\backslash frac$.
** If also Î» = 1:
*** $\backslash mu\; -\; \backslash beta\backslash log\backslash left(\backslash frac\backslash right)\; \backslash sim\; \backslash operatorname(\backslash mu,\; \backslash beta)$, the logistic distribution
*** $\backslash mu\; -\; \backslash beta\backslash log\backslash left(\backslash frac\backslash right)\; \backslash sim\; \backslash operatorname(\backslash mu,\; \backslash beta)$
*** ''Î¼'' âˆ’ Ïƒ log(''X'') ~ generalized extreme value distribution, GEV(Î¼, Ïƒ, 0).
*** Further if $Y\; \backslash sim\; \backslash Gamma\backslash left(\backslash alpha,\; \backslash frac\backslash right)$ then $\backslash sqrt\; \backslash sim\; \backslash operatorname(\backslash alpha,\; \backslash beta)$ (K-distribution)
** If also Î» = 1/2 then ; i.e., ''X'' has a chi-squared distribution with 2 degrees of freedom (statistics), degrees of freedom. Hence: $$\backslash operatorname(\backslash lambda)\; =\; \backslash frac\; \backslash operatorname\backslash left(\backslash frac\; \backslash right)\; \backslash sim\; \backslash frac\; \backslash chi\_2^2\backslash Rightarrow\; \backslash sum\_^n\; \backslash operatorname(\backslash lambda)\; \backslash sim\; \backslash frac\backslash chi\_^2$$
* If $X\; \backslash sim\; \backslash operatorname\backslash left(\backslash frac\backslash right)$ and $Y\; \backslash mid\; X$ ~ Poisson distribution, Poisson(''X'') then $Y\; \backslash sim\; \backslash operatorname\backslash left(\backslash frac\backslash right)$ (geometric distribution
In probability theory
Probability theory 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 expres ...

)
* The Hoyt distribution can be obtained from exponential distribution and arcsine distribution
Other related distributions:
*Hyper-exponential distribution â€“ the distribution whose density is a weighted sum of exponential densities.
*Hypoexponential distribution â€“ the distribution of a general sum of exponential random variables.
*exGaussian distribution â€“ the sum of an exponential distribution and a

_{1}, â€¦, ''x''_{''n''}) drawn from the variable, is:
$$L(\backslash lambda)\; =\; \backslash prod\_^n\backslash lambda\backslash exp(-\backslash lambda\; x\_i)\; =\; \backslash lambda^n\backslash exp\backslash left(-\backslash lambda\; \backslash sum\_^n\; x\_i\backslash right)\; =\; \backslash lambda^n\backslash exp\backslash left(-\backslash lambda\; n\backslash overline\backslash right),$$
where:
$$\backslash overline\; =\; \backslash frac\backslash sum\_^n\; x\_i$$
is the sample mean.
The derivative of the likelihood function's logarithm is:
$$\backslash frac\; \backslash ln\; L(\backslash lambda)\; =\; \backslash frac\; \backslash left(\; n\; \backslash ln\backslash lambda\; -\; \backslash lambda\; n\backslash overline\; \backslash right)\; =\; \backslash frac\; -\; n\backslash overline\backslash \; \backslash begin\; >\; 0,\; \&\; 0\; <\; \backslash lambda\; <\; \backslash frac,\; \backslash \backslash [8pt]\; =\; 0,\; \&\; \backslash lambda\; =\; \backslash frac,\; \backslash \backslash [8pt]\; <\; 0,\; \&\; \backslash lambda\; >\; \backslash frac.\; \backslash end$$
Consequently, the maximum likelihood estimate for the rate parameter is:
$$\backslash widehat\_\backslash text\; =\; \backslash frac\; =\; \backslash frac$$
This is an unbiased estimator of $\backslash lambda,$ although $\backslash overline$ an unbiased MLE estimator of $1/\backslash lambda$ and the distribution mean.
The bias of $\backslash widehat\_\backslash text$ is equal to
$$b\; \backslash equiv\; \backslash operatorname\backslash left[\backslash left(\backslash widehat\_\backslash text\; -\; \backslash lambda\backslash right)\backslash right]\; =\; \backslash frac$$
which yields the Maximum likelihood estimation#Second-order efficiency after correction for bias, bias-corrected maximum likelihood estimator
$$\backslash widehat^*\_\backslash text\; =\; \backslash widehat\_\backslash text\; -\; \backslash widehat\; .$$

gamma distribution
In probability theory
Probability theory 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 express ...

(of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful:
$$\backslash operatorname(\backslash lambda;\; \backslash alpha,\; \backslash beta)\; =\; \backslash frac\; \backslash lambda^\; \backslash exp(-\backslash lambda\backslash beta).$$
The posterior distribution ''p'' can then be expressed in terms of the likelihood function defined above and a gamma prior:
$$\backslash begin\; p(\backslash lambda)\; \&\backslash propto\; L(\backslash lambda)\; \backslash Gamma(\backslash lambda;\; \backslash alpha,\; \backslash beta)\; \backslash \backslash \; \&=\; \backslash lambda^n\; \backslash exp\backslash left(-\backslash lambda\; n\backslash overline\backslash right)\; \backslash frac\; \backslash lambda^\; \backslash exp(-\backslash lambda\; \backslash beta)\; \backslash \backslash \; \&\backslash propto\; \backslash lambda^\; \backslash exp(-\backslash lambda\; \backslash left(\backslash beta\; +\; n\backslash overline\backslash right)).\; \backslash end$$
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:
$$p(\backslash lambda)\; =\; \backslash Gamma(\backslash lambda;\; \backslash alpha\; +\; n,\; \backslash beta\; +\; n\backslash overline).$$
Here the hyperparameter ''Î±'' can be interpreted as the number of prior observations, and ''Î²'' as the sum of the prior observations.
The posterior mean here is:
$$\backslash frac.$$

geometric distribution
In probability theory
Probability theory 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 expres ...

, 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 (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 decays, or the time between clicks of a Geiger counter
* The time it takes before your next telephone call
* 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 mutations on a DNA 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 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 and reliability engineering also make extensive use of the exponential distribution. Because of the ''#Memorylessness, 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

_{''n''+1}, conditioned on the observed samples ''x'' = (''x''_{1}, ..., ''x_{n}'') given by
$$p\_(x\_\; \backslash mid\; x\_1,\; \backslash ldots,\; x\_n)\; =\; \backslash left(\; \backslash frac1\; \backslash right)\; \backslash exp\; \backslash left(\; -\; \backslash frac\; \backslash right)$$
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
$$p\_(x\_\; \backslash mid\; x\_1,\; \backslash ldots,\; x\_n)\; =\; \backslash frac,$$
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 ''Î»'' from the joint likelihood of ''x''_{''n''+1} and ''Î»'' by maximization;
# an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/''Î»'';
# 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, ''Î»''_{0}, and the predictive distribution based on the sample ''x''. The Kullbackâ€“Leibler divergence is a commonly used, parameterisation free measure of the difference between two distributions. Letting Î”(''Î»''_{0}, , ''p'') denote the Kullbackâ€“Leibler divergence between an exponential with rate parameter ''Î»''_{0} and a predictive distribution ''p'' it can be shown that
$$\backslash begin\; \backslash operatorname\_\; \backslash left[\; \backslash Delta(\backslash lambda\_0\backslash parallel\; p\_)\; \backslash right]\; \&=\; \backslash psi(n)\; +\; \backslash frac\; -\; \backslash log(n)\; \backslash \backslash \; \backslash operatorname\_\; \backslash left[\; \backslash Delta(\backslash lambda\_0\backslash parallel\; p\_)\; \backslash right]\; \&=\; \backslash psi(n)\; +\; \backslash frac\; -\; \backslash log(n)\; \backslash end$$
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 Kullbackâ€“Leibler divergence for all sample sizes .

Non-Uniform Random Variate Generation

'. New York: Springer-Verlag. . ''Se

chapter IX

section 2, pp. 392â€“401.'' A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.

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

probability theory
Probability theory 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 expressing it through a set of ...

and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical ...

, the exponential distribution is the probability distribution
In probability theory
Probability theory 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 ...

of the time between events in a Poisson point process
In probability
Probability is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution
In probability theory
Probability theory 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 express ...

. It is the continuous analogue of the geometric distribution
In probability theory
Probability theory 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 expres ...

, and it has the key property of being memoryless
In probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a num ...

. 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
In theory of probability, probability and statistics, an exponential family is a parametric model, parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, based on ...

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
In probability theory
Probability theory 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 ex ...

, binomial distribution
In probability theory
Probability theory 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 ...

, gamma distribution
In probability theory
Probability theory 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 express ...

, Poisson, and many others.
Definitions

Probability density function

Theprobability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...

(pdf) of an exponential distribution is
:$f(x;\backslash lambda)\; =\; \backslash begin\; \backslash lambda\; e^\; \&\; x\; \backslash ge\; 0,\; \backslash \backslash \; 0\; \&\; x\; <\; 0.\; \backslash end$
Here ''Î»'' > 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 is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...

''X'' has this distribution, we write .
The exponential distribution exhibits infinite divisibility
Infinite divisibility arises in different ways in philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy o ...

.
Cumulative distribution function

Thecumulative distribution function
In probability theory
Probability theory 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 exp ...

is given by
:$F(x;\backslash lambda)\; =\; \backslash begin\; 1-e^\; \&\; x\; \backslash ge\; 0,\; \backslash \backslash \; 0\; \&\; x\; <\; 0.\; \backslash end$
Alternative parametrization

The exponential distribution is sometimes parametrized in terms of thescale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family of p ...

, which is also the mean:
$$f(x;\backslash beta)\; =\; \backslash begin\; \backslash frac\; e^\; \&\; x\; \backslash ge\; 0,\; \backslash \backslash \; 0\; \&\; x\; <\; 0.\; \backslash end\; \backslash qquad\backslash qquad\; F(x;\backslash beta)\; =\; \backslash begin\; 1-\; e^\; \&\; x\; \backslash ge\; 0,\; \backslash \backslash \; 0\; \&\; x\; <\; 0.\; \backslash end$$
Properties

Mean, variance, moments, and median

The mean orexpected value
In probability theory
Probability theory 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 exp ...

of an exponentially distributed random variable ''X'' with rate parameter ''Î»'' is given by
$$\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$$
In light of the examples given below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Fred Below (1926â€“1988), American blues drummer
*Fritz von Below (1853â ...

, 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
$$\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$$
so the standard deviation is equal to the mean.
The Moment (mathematics), moments of ''X'', for $n\backslash in\backslash N$ are given by
$$\backslash operatorname\backslash left[X^n\backslash right]\; =\; \backslash frac.$$
The central moments of ''X'', for $n\backslash in\backslash N$ are given by
$$\backslash mu\_n\; =\; \backslash frac\; =\; \backslash frac\backslash sum^n\_\backslash frac.$$
where !''n'' is the Derangement, subfactorial of ''n''
The median of ''X'' is given by
$$\backslash operatorname;\; href="/html/ALL/s/.html"\; ;"title="">$$
where refers to the natural logarithm. Thus the absolute difference between the mean and median is
$$\backslash left,\; \backslash operatorname\backslash left[X\backslash right]\; -\; \backslash operatorname\backslash left[X\backslash right]\backslash \; =\; \backslash frac\; <\; \backslash frac\; =\; \backslash operatorname[X],$$
in accordance with the Chebyshev's inequality#An application: distance between the mean and the median, median-mean inequality.
Memorylessness

An exponentially distributed random variable ''T'' obeys the relation $$\backslash Pr\; \backslash left\; (T\; >\; s\; +\; t\; \backslash mid\; T\; >\; s\; \backslash right\; )\; =\; \backslash Pr(T\; >\; t),\; \backslash qquad\; \backslash forall\; s,\; t\; \backslash ge\; 0.$$ This can be seen by considering the Cumulative distribution function#Complementary cumulative distribution function (tail distribution), complementary cumulative distribution function: $$\backslash begin\; \backslash Pr\backslash left(T\; >\; s\; +\; t\; \backslash mid\; T\; >\; s\backslash right)\; \&=\; \backslash frac\; \backslash \backslash [4pt]\; \&=\; \backslash frac\; \backslash \backslash [4pt]\; \&=\; \backslash frac\; \backslash \backslash [4pt]\; \&=\; e^\; \backslash \backslash [4pt]\; \&=\; \backslash Pr(T\; >\; t).\; \backslash end$$ 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 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 thegeometric distribution
In probability theory
Probability theory 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 expres ...

are memorylessness, 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(''Î»'') is $$F^(p;\backslash lambda)\; =\; \backslash frac,\backslash qquad\; 0\; \backslash le\; p\; <\; 1$$ The quartiles are therefore: *first quartile: ln(4/3)/''Î»'' *median: ln(2)/''Î»'' *third quartile: ln(4)/''Î»'' And as a consequence the interquartile range is ln(3)/''Î»''.Kullbackâ€“Leibler divergence

The directed Kullbackâ€“Leibler divergence in nat (unit), nats of $e^\backslash lambda$ ("approximating" distribution) from $e^$ ('true' distribution) is given by $$\backslash begin\; \backslash Delta(\backslash lambda\_0\; \backslash parallel\; \backslash lambda)\; \&=\; \backslash mathbb\_\backslash left(\; \backslash log\; \backslash frac\backslash right)\backslash \backslash \; \&=\; \backslash mathbb\_\backslash left(\; \backslash log\; \backslash frac\backslash right)\backslash \backslash \; \&=\; \backslash log(\backslash lambda\_0)\; -\; \backslash log(\backslash lambda)\; -\; (\backslash lambda\_0\; -\; \backslash lambda)E\_(x)\backslash \backslash \; \&=\; \backslash log(\backslash lambda\_0)\; -\; \backslash log(\backslash lambda)\; +\; \backslash frac\; -\; 1.\; \backslash end$$Maximum entropy distribution

Among all continuous probability distributions with Support (mathematics)#In probability and measure theory, support and mean ''Î¼'', the exponential distribution with ''Î»'' = 1/''Î¼'' 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''Joint moments of i.i.d. exponential order statistics

Let $X\_1,\; \backslash dotsc,\; X\_n$ be $n$ Independent and identically distributed random variables, independent and identically distributed exponential random variables with rate parameter ''Î»''. Let $X\_,\; \backslash dotsc,\; X\_$ denote the corresponding order statistics. For $i\; <\; j$ , the joint moment $\backslash operatorname\; E\backslash left[X\_\; X\_\backslash right]$ of the order statistics $X\_$ and $X\_$ is given by $$\backslash begin\; \backslash operatorname\; E\backslash left[X\_\; X\_\backslash right]\; \&=\; \backslash sum\_^\backslash frac\; \backslash operatorname\; E\backslash left[X\_\backslash right]\; +\; \backslash operatorname\; E\backslash left[X\_^2\backslash right]\; \backslash \backslash \; \&=\; \backslash sum\_^\backslash frac\backslash sum\_^\backslash frac\; +\; \backslash sum\_^\backslash frac\; +\; \backslash left(\backslash sum\_^\backslash frac\backslash right)^2.\; \backslash end$$ This can be seen by invoking the law of total expectation and the memoryless property: $$\backslash begin\; \backslash operatorname\; E\backslash left[X\_\; X\_\backslash right]\; \&=\; \backslash int\_0^\backslash infty\; \backslash operatorname\; E\backslash left[X\_\; X\_\; \backslash mid\; X\_=x\backslash right]\; f\_(x)\; \backslash ,\; dx\; \backslash \backslash \; \&=\; \backslash int\_^\backslash infty\; x\; \backslash operatorname\; E\backslash left[X\_\; \backslash mid\; X\_\; \backslash geq\; x\backslash right]\; f\_(x)\; \backslash ,\; dx\; \&\&\backslash left(\backslash textrm~X\_\; =\; x\; \backslash implies\; X\_\; \backslash geq\; x\backslash right)\; \backslash \backslash \; \&=\; \backslash int\_^\backslash infty\; x\; \backslash left[\; \backslash operatorname\; E\backslash left[X\_\backslash right]\; +\; x\; \backslash right]\; f\_(x)\; \backslash ,\; dx\; \&\&\backslash left(\backslash text\backslash right)\; \backslash \backslash \; \&=\; \backslash sum\_^\backslash frac\; \backslash operatorname\; E\backslash left[X\_\backslash right]\; +\; \backslash operatorname\; E\backslash left[X\_^2\backslash right].\; \backslash end$$ The first equation follows from the law of total expectation. The second equation exploits the fact that once we condition on $X\_\; =\; x$, it must follow that $X\_\; \backslash geq\; x$. The third equation relies on the memoryless property to replace $\backslash operatorname\; E\backslash left[\; X\_\; \backslash mid\; X\_\; \backslash geq\; x\backslash right]$ with $\backslash operatorname\; E\backslash left[X\_\backslash right]\; +\; x$.Sum of two independent exponential random variables

The probability distribution function (PDF) of a sum of two independent random variables is the convolution of probability distributions, convolution of their individual PDFs. If $X\_1$ and $X\_2$ are independent exponential random variables with respective rate parameters $\backslash lambda\_1$ and $\backslash lambda\_2,$ then the probability density of $Z=X\_1+X\_2$ is given by $$\backslash begin\; f\_Z(z)\; \&=\; \backslash int\_^\backslash infty\; f\_(x\_1)\; f\_(z\; -\; x\_1)\backslash ,dx\_1\backslash \backslash \; \&=\; \backslash int\_0^z\; \backslash lambda\_1\; e^\; \backslash lambda\_2\; e^\; \backslash ,\; dx\_1\; \backslash \backslash \; \&=\; \backslash lambda\_1\; \backslash lambda\_2\; e^\; \backslash int\_0^z\; e^\backslash ,dx\_1\; \backslash \backslash \; \&=\; \backslash begin\; \backslash dfrac\; \backslash left(e^\; -\; e^\backslash right)\; \&\; \backslash text\; \backslash lambda\_1\; \backslash neq\; \backslash lambda\_2\; \backslash \backslash [4\; pt]\; \backslash lambda^2\; z\; e^\; \&\; \backslash text\; \backslash lambda\_1\; =\; \backslash lambda\_2\; =\; \backslash lambda.\; \backslash end\; \backslash end$$ The entropy of this distribution is available in closed form: assuming $\backslash lambda\_1\; >\; \backslash lambda\_2$ (without loss of generality), then $$\backslash begin\; H(Z)\; \&=\; 1\; +\; \backslash gamma\; +\; \backslash ln\; \backslash left(\; \backslash frac\; \backslash right)\; +\; \backslash psi\; \backslash left(\; \backslash frac\; \backslash right)\; ,\; \backslash end$$ where $\backslash gamma$ is the Euler-Mascheroni constant, and $\backslash psi(\backslash cdot)$ is the digamma function. In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter $\backslash lambda,$ which in turn is a special case ofgamma distribution
In probability theory
Probability theory 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 express ...

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Related distributions

* If $X\; \backslash sim\; \backslash operatorname\backslash left(\backslash mu,\; \backslash beta^\backslash right)$ then , ''X'' âˆ’ Î¼, ~ Exp(Î²). * If ''X'' ~ Pareto(1, Î») then log(''X'') ~ Exp(Î»). * If ''X'' ~ skew-logistic distribution, SkewLogistic(Î¸), then $\backslash log\backslash left(1\; +\; e^\backslash right)\; \backslash sim\; \backslash operatorname(\backslash theta)$. * If ''Xgeometric distribution
In probability theory
Probability theory 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 expres ...

on 0,1,2,3,...
** $\backslash lceil\; X\backslash rceil\; \backslash sim\; \backslash operatorname\backslash left(1-e^\backslash right)$, a normal distribution
In probability theory
Probability theory 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 ex ...

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Statistical inference

Below, suppose random variable ''X'' is exponentially distributed with rate parameter Î», and $x\_1,\; \backslash dotsc,\; x\_n$ are ''n'' independent samples from ''X'', with sample mean $\backslash bar$.Parameter estimation

The maximum likelihood estimator for Î» is constructed as follows: The likelihood function for Î», given an independent identically-distributed random variables, independent and identically distributed sample ''x'' = (''x''Approximate minimizer of expected squared error

Assume you have at least three samples. If we seek a minimizer of expected mean squared error (see also: Biasâ€“variance tradeoff) that is similar to the maximum likelihood estimate (i.e. a multiplicative correction to the likelihood estimate) we have: $$\backslash widehat\; =\; \backslash left(\backslash frac\backslash right)\; \backslash left(\backslash frac\backslash right)\; =\; \backslash frac$$ This is derived from the mean and variance of the inverse-gamma distribution: $\backslash mbox(n,\; \backslash lambda)$.Fisher information

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

The 100(1 âˆ’ Î±)% confidence interval for the rate parameter of an exponential distribution is given by: $$\backslash frac<\; \backslash frac\; <\; \backslash frac$$ which is also equal to: $$\backslash frac\; <\; \backslash frac\; <\; \backslash frac$$ where is the percentile of the chi squared distribution with ''v'' degrees of freedom (statistics), 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 distribution. This approximation gives the following values for a 95% confidence interval: $$\backslash begin\; \backslash lambda\_\backslash text\; \&=\; \backslash widehat\backslash left(1\; -\; \backslash frac\backslash right)\; \backslash \backslash \; \backslash lambda\_\backslash text\; \&=\; \backslash widehat\backslash left(1\; +\; \backslash frac\backslash right)\; \backslash end$$ This approximation may be acceptable for samples containing at least 15 to 20 elements.Bayesian inference

The conjugate prior for the exponential distribution is theOccurrence and 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 thebinomial distribution
In probability theory
Probability theory 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 ...

. 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 Predictive methods for surgery duration, no typical work-contnet (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 ''Î»'' 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''Computational methods

Generating exponential variates

A conceptually very simple method for generating exponential random variate, variates is based on inverse transform sampling method, inverse transform sampling: Given a random variate ''U'' drawn from the uniform distribution (continuous), uniform distribution on the unit interval , the variate $$T\; =\; F^(U)$$ has an exponential distribution, where ''F'' is the quantile function, defined by $$F^(p)=\backslash frac.$$ Moreover, if ''U'' is uniform on (0, 1), then so is 1 âˆ’ ''U''. This means one can generate exponential variates as follows: $$T\; =\; \backslash frac.$$ Other methods for generating exponential variates are discussed by KnuthDonald Knuth, Donald E. Knuth (1998). ''The Art of Computer Programming'', volume 2: ''Seminumerical Algorithms'', 3rd edn. Boston: Addisonâ€“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. 392â€“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 â€“ an application of exponential distribution to particle detector analysis. * Laplace distribution, or the "double exponential distribution". * Relationships among probability distributions * Marshallâ€“Olkin exponential distributionReferences

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