Exponential distribution

exponentialexponentially distributedexponentiallyNegative exponentialexponential random variablenegative exponential distributionConfidence interval for mean of the exponential distributionconstant failure rateexponential density functionexponential model
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.wikipedia
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Gamma distribution

gammagamma distributedGamma variate
It is a particular case of the gamma distribution. The exponential distribution is not the same as the class of exponential families 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, and many others. In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter \lambda, which in turn is a special case of gamma distribution.
The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution – or for that matter, the β of the gamma distribution itself.

Exponential family

exponential familiesnatural parameternatural parameters
The exponential distribution is not the same as the class of exponential families 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, and many others.
The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families.

Memorylessness

memorylessmemory-less
It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless.
Only two kinds of distributions are memoryless: exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers.

Variance

sample variancepopulation variancevariability
The variance of X is given by
The exponential distribution with parameter λ is a continuous distribution whose probability density function is given by

Poisson distribution

PoissonPoisson-distributedPoissonian
The exponential distribution is not the same as the class of exponential families 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, and many others.

Quantile function

quantileinverse distribution functionnormal quantile function
The quantile function (inverse cumulative distribution function) for Exp is
For example, the cumulative distribution function of Exponential (i.e.

Geometric distribution

geometricgeometrically distributed geometrically distributed
It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless.
* Like its continuous analogue (the exponential distribution), the geometric distribution is memoryless.

Erlang distribution

ErlangErlang-C
In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter \lambda, which in turn is a special case of gamma distribution.
The Erlang distribution with shape parameter k=1 simplifies to the exponential distribution.

Scale parameter

scalerate parameterestimation
The exponential distribution is sometimes parametrized in terms of the scale parameter
So for example the exponential distribution with scale parameter β and probability density

Median

averagesample medianmedian-unbiased estimator
The median of X is given by

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Laplace distribution

Laplacedouble exponentialLaplace distributed
It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together back-to-back, although the term is also sometimes used to refer to the Gumbel distribution.

Probability theory

theory of probabilityprobabilityprobability theorist
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.
Important continuous distributions include the continuous uniform, normal, exponential, gamma and beta distributions.

Failure rate

hazard functionMean Distance Between FailuresDecreasing failure rate
The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate.
which is based on the exponential density function.

Maximum entropy probability distribution

maximum entropymaximum entropy distributionlargest 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.
The exponential distribution, for which the density function is

Weibull distribution

WeibullWeibull chartRosin-Rammler distribution
The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and ).

Random variable

random variablesrandom variationrandom
If a random variable X has this distribution, we write X ~ Exp.
which is the cumulative distribution function (CDF) of an exponential distribution.

Normal distribution

normally distributedGaussian distributionnormal
The exponential distribution is not the same as the class of exponential families 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, and many others.

Differential entropy

differential entropiesdifferential Shannon informationentropy
Among all continuous probability distributions with support [0, ∞) and mean μ, the exponential distribution with λ = 1/μ has the largest differential entropy.
Let X be an exponentially distributed random variable with parameter \lambda, that is, with probability density function

Order statistic

order statisticsk'th-smallest of n itemsordered
Let denote the corresponding order statistics.
For random samples from an exponential distribution with parameter λ, the order statistics X (i) for i = 1,2,3, ..., n each have distribution

Exponentially modified Gaussian distribution

ExGaussian distributionEMG distributionExponentially modified Gaussian
In probability theory, an exponentially modified Gaussian (EMG) distribution (exGaussian distribution) describes the sum of independent normal and exponential random variables.

Lomax distribution

Lomax
This is the distribution of the ratio of two independent and identically distributed random variables with exponential distributions.

Hyperexponential distribution

Hyper-exponential distributionhyperexponential distributionshyperexponentially distributed
where each Y i is an exponentially distributed random variable with rate parameter λ i, and p i is the probability that X will take on the form of the exponential distribution with rate λ i.

Chi-squared distribution

chi-squaredchi-square distributionchi square distribution
percentile of the chi squared distribution with v degrees of freedom, n is the number of observations of inter-arrival times in the sample, and x-bar is the sample average.
Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-square distribution is not as often applied in the direct modeling of natural phenomena.

Beta distribution

beta betabeta of the first kind
"1/p" is Jeffreys' (1946) invariant prior for the exponential distribution, not for the Bernoulli or binomial distributions).