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 variableX has this distribution, we write X ~ Exp(λ).
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 first equation follows from the law of total expectation.
The second equation exploits the fact that once we condition on , it must follow that .
The third equation relies on the memoryless property to replace with .
Sum of two independent exponential random variables
The probability distribution function (PDF) of a sum of two independent random variables is the convolution of their individual PDFs. If and are independent exponential random variables with respective rate parameters and then the probability density of is given by
The entropy of this distribution is available in closed form: assuming (without loss of generality), then
Assume you have at least three samples. If we seek a minimizer of expected mean squared error (see also: Bias–variance tradeoff) that is similar to the maximum likelihood estimate (i.e. a multiplicative correction to the likelihood estimate) we have:
This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter .
The 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by:
which is also equal to:
where χ2 p,v is the 100(p)percentile of the chi squared distribution with vdegrees 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 χ2 p,v distribution. This approximation gives the following values for a 95% confidence interval:
This approximation may be acceptable for samples containing at least 15 to 20 elements.
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:
The posterior distributionp 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 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:
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.
The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.
In real-world scenarios, the assumption of a constant rate (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 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 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.
Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq
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
a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of xn+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, λ0, and the predictive distribution based on the sample x. The Kullback–Leibler divergence is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(λ0||p) denote the Kullback–Leibler divergence between an exponential with rate parameter λ0 and a predictive distribution p it can be shown that
where the expectation is taken with respect to the exponential distribution with rate parameter λ0 ∈ (0, ∞), and ψ( · ) is the digamma function. It is clear that the CNML predictive distribution is strictly superior to the maximum likelihood plug-in distribution in terms of average Kullback–Leibler divergence for all sample sizes n > 0.
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