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Encyclopedia > Exponential distribution
Exponential
Probability density function
Probability density function
Cumulative distribution function
Cumulative distribution function
Parameters lambda > 0 , rate or inverse scale (real)
Support [0,infty)!
Probability density function (pdf) λe − λx
Cumulative distribution function (cdf) 1 − e − λx
Mean lambda^{-1},
Median ln(2)/lambda,
Mode 0,
Variance lambda^{-2},
Skewness 2,
Excess kurtosis 6,
Entropy 1 - ln(lambda),
Moment-generating function (mgf) left(1 - frac{t}{lambda}right)^{-1},
Characteristic function left(1 - frac{it}{lambda}right)^{-1},

In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. They are often used to model the time between independent events that happen at a constant average rate. Download high resolution version (1300x975, 118 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x975, 119 KB) See the image on the commons for gnuplot source. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability theory is a branch of mathematics concerned with analysis of random phenomena. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

Contents

Characterization

Probability density function

The probability density function (pdf) of an exponential distribution has the form In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

f(x;lambda) = left{begin{matrix} lambda e^{-lambda x} &,; x ge 0,  0 &,; x < 0. end{matrix}right.

where λ > 0 is a parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0,∞). If a random variable X has this distribution, we write X ~ Exponential(λ). A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...


Cumulative distribution function

The cumulative distribution function is given by In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

F(x;lambda) = left{begin{matrix} 1-e^{-lambda x}&,; x ge 0,  0 &,; x < 0. end{matrix}right.

Alternate parameterization

A commonly used alternate parameterization is to define the probability density function (pdf) of an exponential distribution as In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

f(x;beta) = left{begin{matrix} frac{1}{beta} e^{-x/beta} &,; x ge 0,  0 &,; x < 0. end{matrix}right.

where β > 0 is a scale parameter of the distribution and is the reciprocal of the rate parameter, λ, defined above. In this specification, β is a survival parameter in the sense that if a random variable X is the duration of time that a given biological or mechanical system M manages to survive and X ~ Exponential(β) then mathbb{E}[X] = beta. That is to say, the expected duration of survival of M is β units of time. In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... The reciprocal function: y = 1/x. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...


This alternate specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. We shall not assume this alternate specification. Unfortunately this gives rise to a notational ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "X ~ 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. The term notation can be used in several contexts. ...


Occurrence and applications

The exponential distribution is used to model Poisson processes, which are situations in which an object initially in state A can change to state B with constant probability per unit time λ. The time at which the state actually changes is described by an exponential random variable with parameter λ. Therefore, the integral from 0 to T over f is the probability that the object is in state B at time T. It has been suggested that this article be split into multiple articles. ...


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 probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... 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, called success and failure. ...


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 beeps of a geiger counter;
  • the number of dice rolls needed until you roll a six 11 times in a row;.
  • 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 distance: This article or section does not cite any references or sources. ...

  • the distance between mutations on a DNA strand;
  • the distance between roadkill on a given street;

In queuing theory, the inter-arrival times (i.e. the times between customers entering the system) are often modeled as exponentially distributed variables. The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the Erlang distribution (which is a sum of several independent exponentially distributed variables). It has been suggested that mutant be merged into this article or section. ... The structure of part of a DNA double helix Deoxyribonucleic acid (DNA) is a nucleic acid that contains the genetic instructions for the development and function of living organisms. ... Queueing theory (spelled queuing theory in the United States) is the mathematical study of waiting lines (or queues). ... The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. ...


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. Reliability theory developed apart from the mainstream of probability and statistics, and was used originally as a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. ... Reliability engineering is the discipline of ensuring that a system will be reliable when operated in a specified manner. ... Exponential failure density functions A failure rate is the average frequency with which something fails. ... In reliability theory, the bathtub curve is the phenomenon that the fraction of products failing in a given timespan is usually high early in the lifecycle, low in the middle, and rising strongly towards the end. ... Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. ...


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. This is a consequence of the entropy property mentioned below. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... This article or section does not cite its references or sources. ... Fig. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ...


Properties

Mean and variance

The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

mathrm{E}[X] = frac{1}{lambda}. !

In light of the examples given above, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.


The variance of X is given by In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

mathrm{Var}[X] = frac{1}{lambda^2}. !

Memorylessness

An important property of the exponential distribution is that it is memoryless. This means that if a random variable T is exponentially distributed, its conditional probability obeys In probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. ... This article defines some terms which characterize probability distributions of two or more variables. ...

P(T > s + t; |; T > s) = P(T > t) ;; hbox{for all} s, t ge 0.

This says that the conditional probability 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 no different from the initial probability that we need to wait more than 10 seconds for the first arrival. This is often misunderstood by students taking courses on probability: the fact that P(T > 40 | T > 30) = P(T > 10) does not mean that the events T > 40 and T > 30 are independent. To summarize: "memorylessness" of the probability distribution of the waiting time T until the first arrival means This article defines some terms which characterize probability distributions of two or more variables. ...

mathrm{(Right)} P(T>40 mid T>30)=P(T>10).

It does not mean

mathrm{(Wrong)} P(T>40 mid T>30)=P(T>40).

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


The exponential distributions and the geometric distributions are the only memoryless probability distributions. In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before...


The exponential distribution also has a constant hazard function. Exponential failure density functions A failure rate is the average frequency with which something fails. ...


Quartiles

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

F^{-1}(p;lambda) = frac{-ln(1-p)}{lambda}, !

for 0 le p < 1. The quartiles are therefore: In descriptive statistics, a quartile is any of the three values which divide the sorted data set into four equal parts, so that each part represents 1/4th of the sample or population. ...

first quartile 
ln(4/3)/lambda,
median 
ln(2)/lambda,
third quartile 
ln(4)/lambda,

In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ...

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. In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ...


Parameter estimation

Suppose you know that a given variable is exponentially distributed and you want to estimate the rate parameter λ.


Maximum likelihood

The likelihood function for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from your variable, is Look up likelihood in Wiktionary, the free dictionary. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

L(lambda) = prod_{i=1}^n lambda , exp(-lambda x_i) = lambda^n , exp!left(!-lambda sum_{i=1}^n x_iright)=lambda^nexpleft(-lambda n overline{x}right),

where

overline{x}={1 over n}sum_{i=1}^n x_i

is the sample mean.


The derivative of the likelihood function's logarithm is

frac{mathrm{d}}{mathrm{d}lambda} ln L(lambda) = frac{mathrm{d}}{mathrm{d}lambda} left( n ln(lambda) - lambda noverline{x} right) = {n over lambda}-noverline{x} left{begin{matrix} > 0 & mbox{if} 0 < lambda < 1/overline{x},   = 0 & mbox{if} lambda = 1/overline{x},   < 0 & mbox{if} lambda > 1/overline{x}. end{matrix}right.

Consequently the maximum likelihood estimate for the rate parameter is Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ...

widehat{lambda} = frac1{overline{x}}.

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 pdf is useful: In Bayesian probability theory, a class of prior probability distributions p(θ) is said to be conjugate to a class of likelihood functions p(x|θ) if the resulting posterior distributions p(θ|x) are in the same family as p(θ). For example, the Gaussian family is conjugate to itself (or self-conjugate... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ...

mathrm{Gamma}(lambda ,;, alpha, beta) = frac{beta^{alpha}}{Gamma(alpha)} , lambda^{alpha-1} , exp(-lambda,beta). !

The posterior distribution p can then be expressed in terms of the likelihood function defined above and a gamma prior: In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. ...

p(lambda) propto L(lambda) times mathrm{Gamma}(lambda ,;, alpha, beta)
= lambda^n , exp(-lambda,noverline{x}) times frac{beta^{alpha}}{Gamma(alpha)} , lambda^{alpha-1} , exp(-lambda,beta)
propto lambda^{(alpha+n)-1} , exp(-lambda,(beta + noverline{x})).

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) = mathrm{Gamma}(lambda ,;, alpha + n, beta + n overline{x}).

Here the parameter α can be interpreted as the number of prior observations, and β as the sum of the prior observations.


Generating exponential variates

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval (0,1), the variate In probability theory, a random variable is a measurable function from a probability space to a measurable space of values the variable can take on. ... The inverse transform sampling method is a method of sampling a number at random from any probability distribution given its cumulative distribution function (cdf). ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ...

T = F^{-1}(U) !

has an exponential distribution, where F − 1 is the quantile function, defined by

F^{-1}(p)=frac{-ln(1-p)}{lambda}. !

Moreover, if U is uniform on (0;1), then so is 1 − U. This means one can generate exponential variates as follows:

T = frac{-ln U}{lambda}. !

Other methods for generating exponential variates are discussed by Knuth[1] and Devroye[2].


The ziggurat algorithm is a fast method for generating exponential variates. The ziggurat algorithm is a rejection sampling algorithm which generates normally-distributed pseudo-random numbers. ...


Related distributions

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... This article needs cleanup. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ... This article or section is in need of attention from an expert on the subject. ... In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ... In probability theory and statistics, the chi-square distribution (also chi-squared or χ2  distribution) is one of the theoretical probability distributions most widely used in inferential statistics, i. ...

References

  1. ^ Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn. Boston: Addison-Wesley. ISBN 0-201-89684-2. See section 3.4.1, p. 133.
  2. ^ Luc Devroye (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. ISBN 0-387-96305-7. See chapter IX, section 2, pp. 392–401.
Image:Bvn-small.png Probability distributionsview  talk  edit ]
Univariate Multivariate
Discrete: BenfordBernoullibinomialBoltzmanncategoricalcompound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniform • Yule-Simon • zetaZipf • Zipf-Mandelbrot Ewensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta function • Coxian • Erlangexponentialexponential powerFfading • Fisher's z • Fisher-Tippett • Gammageneralized extreme valuegeneralized hyperbolicgeneralized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussianinverse gamma (scaled inverse gamma) • KumaraswamyLandauLaplaceLévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speednormal (Gaussian) • normal-gamma • normal inverse Gaussian • ParetoPearson • phase-type • polarraised cosineRayleigh • relativistic Breit-Wigner • Riceshifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigtvon MisesWeibullWigner semicircleWilks' lambda Dirichlet • inverse-Wishart • Kentmatrix normalmultivariate normalmultivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantorconditionalequilibriumexponential family • infinitely divisible • location-scale family • marginalmaximum entropyposteriorprior • quasi • samplingsingular

  Results from FactBites:
 
Exponential distribution - Wikipedia, the free encyclopedia (1380 words)
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.
An important property of the exponential distribution is that it is memoryless.
The conjugate prior for the exponential distribution is the gamma distribution (of which the exponential distribution is a special case).
Exponential decay - Wikipedia, the free encyclopedia (1091 words)
Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the law of large numbers holds.
In a sample of a radionuclide that undergoes radioactive decay to a different state, the number of atoms in the original state follows exponential decay as long as the remaining number of atoms is large.
If an object at one temperature is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay (in the limit of slow processes; equivalent to "good" heat conduction inside the object, so that its temperature remains relatively uniform throught its volume).
  More results at FactBites »

 
 

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