Exponential distribution

# Exponential distribution

Overview
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the exponential distribution (a.k.a. negative exponential distribution) is a family of continuous probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s. It describes the time between events in a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

, i.e. a process in which events occur continuously and independently
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

at a constant average rate.
Discussion

Encyclopedia
In probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the exponential distribution (a.k.a. negative exponential distribution) is a family of continuous probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s. It describes the time between events in a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

, i.e. a process in which events occur continuously and independently
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

at a constant average rate.

Note that the exponential distribution is not the same as the class of exponential families
Exponential family
In probability and statistics, an exponential family is an important class of probability distributions sharing a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential...

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
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 and/or space if these events occur with a known average rate and independently of the time since...

, and many others.

### Probability density function

The probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

(pdf) of an exponential distribution is

Alternatively, this can be defined using the Heaviside step function
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

, 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
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

X has this distribution, we write X ~ Exp(λ).

The exponential distribution exhibits infinite divisibility
Infinite divisibility (probability)
The concepts of infinite divisibility and the decomposition of distributions arise in probability and statistics in relation to seeking families of probability distributions that might be a natural choice in certain applications, in the same way that the normal distribution is...

.

### Cumulative distribution function

The cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

is given by

Alternatively, this can be defined using the Heaviside step function
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

H(x).

### Alternative parameterization

A commonly used alternative parameterization is to define the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

(pdf) of an exponential distribution as

where β > 0 is a scale parameter
Scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

of the distribution and is the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of the rate parameter, λ, defined above. In this specification, β is a survival parameter in the sense that if a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

X is the duration of time that a given biological or mechanical system manages to survive and X ~ Exponential(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 (which might be modelled using an exponential distribution) has a mean of β = λ−1.

The alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. This alternative specification is not used here. Unfortunately this gives rise to a notational
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...

ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "X ~ Exponential(λ)", since either the notation in the previous (using λ) or the notation in this section (here, using β to avoid confusion) could be intended.

### Mean, variance, and median

The mean or expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

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
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

of X is given by

The median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

of X is given by

where ln refers to the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

. Thus the absolute difference
Absolute difference
The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y...

between the mean and median is

in accordance with the median-mean inequality.

### Memorylessness

An important property of the exponential distribution is that it is memoryless
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

. This means that if a random variable T is exponentially distributed, its conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

obeys

This says that the conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

that we need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet happened after 30 seconds, is equal to the initial probability that we need to wait more than 10 seconds for the first arrival. So, if we waited for 30 seconds and the first arrival didn't happen (T > 30), probability that we'll need to wait another 10 seconds for the first arrival (T > 30 + 10) is the same as the initial probability that we need to wait more than 10 seconds for the first arrival (T > 10). Students taking courses on probability often misunderstand this idea: the fact that Pr(T > 40 | T > 30) = Pr(T > 10) does not mean that the events T > 40 and T > 30 are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

.

To summarize: "memorylessness" of the probability distribution of the waiting time T until the first arrival means

It does not mean

(That would be independence. These two events are not independent.)

The exponential distributions and the geometric distributions 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 rate
Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. It is often denoted by the Greek letter λ and is important in reliability engineering....

.

### Quantiles

The quantile function
Quantile function
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...

(inverse cumulative distribution function) for Exponential(λ) is

for 0 ≤ p < 1.
The quartile
Quartile
In descriptive statistics, the quartiles of a set of values are the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled...

s are therefore:

first quartile : ln(4/3)/λ
median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

: ln(2)/λ
third quartile : ln(4)/λ

### Kullback–Leibler divergence

The directed Kullback–Leibler divergence
Kullback–Leibler divergence
In probability theory and information theory, the Kullback–Leibler divergence is a non-symmetric measure of the difference between two probability distributions P and Q...

between Exp(λ0) ('true' distribution) and Exp(λ) ('approximating' distribution) is given by

### Maximum entropy distribution

Among all continuous probability distributions with support [0,∞) and mean μ, the exponential distribution with λ = 1/μ has the largest entropy. Alternatively, it is the maximum entropy probability distribution
Maximum entropy probability distribution
In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is at least as great as that of all other members of a specified class of distributions....

for a random variate X 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
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...

for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from the variable, is

where

is the sample mean.

The derivative of the likelihood function's logarithm is

Consequently the maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....

estimate for the rate parameter is

While this estimate is the most likely reconstruction of the true parameter λ, it is only an estimate, and as such, one can imagine that the more data points are available the better the estimate will be. It so happens that one can compute an exact confidence interval
Confidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...

– that is, a confidence interval that is valid for all number of samples, not just large ones. The 100(1 − α)% exact confidence interval for this estimate is given by

which is also equal to:

where is the MLE estimate, is the true value of the parameter, and is the percentile
Percentile
In statistics, a percentile is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value below which 20 percent of the observations may be found...

of the chi squared distribution with degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

.

### Bayesian inference

The conjugate prior
Conjugate prior
In Bayesian probability theory, if the posterior distributions p are in the same family as the prior probability distribution p, the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood...

for the exponential distribution is the gamma distribution (of which the exponential distribution is a special case). The following parameterization of the gamma pdf 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.

### Confidence interval

A simple and rapid method to calculate an approximate confidence interval for the estimation of λ is based on the application of the central limit theorem. This method provides a good approximation of the confidence interval limits, for samples containing at least 15 – 20 elements. Denoting by N the sample size, the upper and lower limits of the 95% confidence interval are given by:

## Generating exponential variates

A conceptually very simple method for generating exponential variate
Random variate
A random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable would have different values. Random variates are used when simulating processes driven by random influences...

s is based on inverse transform sampling
Inverse transform sampling method
Inverse transform sampling, also known as the inverse probability integral transform or inverse transformation method or Smirnov transform or even golden rule, is a basic method for pseudo-random number sampling, i.e. for generating sample numbers at random from any probability distribution given...

: Given a random variate U drawn from the uniform distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

on the unit interval (0, 1), the variate

has an exponential distribution, where F −1 is the quantile function
Quantile function
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...

, defined by

Moreover, if U is uniform on (0, 1), 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
Ziggurat algorithm
The ziggurat algorithm is an algorithm for pseudo-random number sampling. Belonging to the class of rejection sampling algorithms, it relies on an underlying source of uniformly-distributed random numbers, typically from a pseudo-random number generator, as well as precomputed tables. The...

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 then
• If and then
• If then
• The Benktander Weibull distribution reduces to a truncated exponential distribution
• If then (Benktander Weibull distribution)
• The exponential distribution is a limit of a scaled beta distribution:
• If then (Erlang distribution)
• If then (Generalized extreme value distribution)
• If then (gamma distribution)
• If and then (Laplace distribution)
• If and then
• If then
• If then (logistic distribution)
• If and then (logistic distribution)
• If then (Pareto distribution)
• If then
• Exponential distribution is a special case of type 3 Pearson distribution
Pearson distribution
The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics.- History :...

• If then (power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

)
• If then (Rayleigh distribution)
• If then (Weibull distribution)
• If then (Weibull distribution)
• If (Uniform distribution (continuous)
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

) then
• If (Poisson distribution
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 and/or space if these events occur with a known average rate and independently of the time since...

) where then (geometric distribution)
• If and then (K-distribution
K-distribution
The K-distribution is a probability distribution that arises as the consequence of a statistical or probabilistic model used in Synthetic Aperture Radar imagery...

)
• The Hoyt distribution  can be obtained from Exponential distribution and Arcsine distribution
• If and then
• If and then , i.e. Y has a Gumbel distribution, if and ., i.e. X has a chi-squared distribution with 2 degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

, if ., then : see skew-logistic distribution.
• Let and be independent. Then has probability density function . This can be used to obtain a confidence interval
Confidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...

for .

Other related distributions:
• Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities.
• Hypoexponential distribution
Hypoexponential distribution
In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes...

– the distribution of a general sum of exponential random variables.
• exGaussian distribution – the sum of an exponential distribution and a normal distribution.

### Occurrence of events

The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trial
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"....

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
Particle decay
Particle decay is the spontaneous process of one elementary particle transforming into other elementary particles. During this process, an elementary particle becomes a different particle with less mass and an intermediate particle such as W boson in muon decay. The intermediate particle then...

s, or the time between clicks of a geiger counter
Geiger counter
A Geiger counter, also called a Geiger–Müller counter, is a type of particle detector that measures ionizing radiation. They detect the emission of nuclear radiation: alpha particles, beta particles or gamma rays. A Geiger counter detects radiation by ionization produced in a low-pressure gas in a...

• 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 mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...

s on a DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

Roadkill is an animal or animals that have been struck and killed by motor vehicles. In the United States of America, removal and disposal of animals struck by motor vehicles is usually the responsibility of the state's state trooper association or department of transportation.-History:During the...

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 inter-arrival of customers for instance in a system is typically modeled by the Poisson distribution
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 and/or space if these events occur with a known average rate and independently of the time since...

in most management science textbooks.) The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).
Reliability theory
Reliability theory
Reliability theory describes the probability of a system completing its expected function during an interval of time. It is the basis of reliability engineering, which is an area of study focused on optimizing the reliability, or probability of successful functioning, of systems, such as airplanes,...

and reliability engineering
Reliability engineering
Reliability engineering is an engineering field, that deals with the study, evaluation, and life-cycle management of reliability: the ability of a system or component to perform its required functions under stated conditions for a specified period of time. It is often measured as a probability of...

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
Bathtub curve
The bathtub curve is widely used in reliability engineering. It describes a particular form of the hazard function which comprises three parts:*The first part is a decreasing failure rate, known as early failures....

used in reliability theory. It is also very convenient because it is so easy to add failure rate
Failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. It is often denoted by the Greek letter λ and is important in reliability engineering....

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
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, if you observe a gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

at a fixed temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

and pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

in a uniform gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...

, the heights of the various molecules also follow an approximate exponential distribution. This is a consequence of the entropy property mentioned below.

In hydrology
Hydrology
Hydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...

, 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
Cumulative frequency analysis
Cumulative frequency analysis is the applcation of estimation theory to exceedance probability . The complement, the non-exceedance probability concerns the frequency of occurrence of values of a phenomenon staying below a reference value. The phenomenon may be time or space dependent...

.

### 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 = (x1, ..., xn) 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
Confidence distribution
In statistics, the concept of a confidence distribution has often been loosely referred to as a distribution function on the parameter space that can represent confidence intervals of all levels for a parameter of interest...

, obtained from the distribution of the pivotal quantity ;
(2) a profile predictive likelihood, obtained by eliminating the parameter from the joint likelihood of and by maximization;
(3) an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior
Jeffreys prior
In Bayesian probability, the Jeffreys prior, named after Harold Jeffreys, is a non-informative prior distribution on parameter space that is proportional to the square root of the determinant of the Fisher information:...

;
and (4) 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, λ0, and the predictive distribution based on the sample x. The Kullback–Leibler divergence
Kullback–Leibler divergence
In probability theory and information theory, the Kullback–Leibler divergence is a non-symmetric measure of the difference between two probability distributions P and Q...

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 , 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 .