Exponential distribution

exponentialexponentially distributedexponentiallyexponential random variablenegative exponentialnegative exponential distributionConfidence interval for mean of the exponential distributionconstant failure rateexponential density functionexponential distributions
In probability theory and statistics, the exponential distribution (also known as the negative exponential distribution) is the probability distribution that describes 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 distributed random variables
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. If X i ~ Exp then the sum X 1 + ... + X k = \sum_i X_i ~ Erlang(k, λ) which is just a Gamma(k, λ −1 ) (in (k, θ) parametrization) or Gamma(k, λ) (in parametrization) with an integer shape parameter k.
The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution.

Exponential family

exponential familiesnatural parametersnatural statistics
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.
exponential

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 \lambda is a continuous distribution whose support is the semi-infinite interval [0, \infty).

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.
If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/λ.

Mean

mean valuepopulation meanaverage
where β > 0 is mean, standard deviation, and scale parameter of the distribution, the reciprocal of the rate parameter, λ, defined above.
While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions.

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.

Scale parameter

scalerate parameterestimation
where β > 0 is mean, standard deviation, and scale parameter of the distribution, the reciprocal of the rate parameter, λ, defined above.
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
The median of an exponential distribution with rate parameter λ is the natural logarithm of 2 divided by the rate parameter: λ −1 ln 2.

Geometric distribution

geometricgeometrically distributedgeometric random variable
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. That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is the only memoryless discrete distribution.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
In probability theory and statistics, the exponential distribution (also known as the negative exponential distribution) is the probability distribution that describes 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.
Pareto distribution, for a single such quantity whose log is exponentially distributed; the prototypical power law distribution

Probability theory

theory of probabilityprobabilityprobability theorist
In probability theory and statistics, the exponential distribution (also known as the negative exponential distribution) is the probability distribution that describes 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 failurescomponent 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

Erlang distribution

ErlangErlang(''k'', ''λ'')
If X i ~ Exp then the sum X 1 + ... + X k = \sum_i X_i ~ Erlang(k, λ) which is just a Gamma(k, λ −1 ) (in (k, θ) parametrization) or Gamma(k, λ) (in parametrization) with an integer shape parameter k.
The Erlang distribution with shape parameter k=1 simplifies to the exponential distribution.

Random variable

random variablesrandom variationrandom
The distribution is supported on the interval [0, ∞). 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 distributednormalGaussian
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. exGaussian distribution – the sum of an exponential distribution and a normal distribution.
will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector (X, Y) the squared norm X 2 + Y 2 will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.

Laplace distribution

Laplacedouble exponentialLaplace distributed
If X ~ Exp and Y ~ Exp then λX − νY ~ Laplace(0, 1).
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.

Chi-squared distribution

chi-squaredchi-square distributionchi square
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-squared distribution is not as often applied in the direct modeling of natural phenomena.

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 statisticsorderedth-smallest of items
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

Weibull distribution

WeibullRosin-Rammler distributionWeibull curve
If X ~ Exp then (Weibull 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 ).

Exponentially modified Gaussian distribution

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

Uniform distribution (continuous)

uniform distributionuniformuniformly distributed
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
If X has a standard uniform distribution, then by the inverse transform sampling method, Y = − λ −1 ln(X) has an exponential distribution with (rate) parameter λ.

Hyperexponential distribution

hyper-exponential distributionhyperexponential distributionshyperexponentially distributed
Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities.
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 . It is named the hyperexponential distribution since its coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and the hypoexponential distribution, which has a coefficient of variation smaller than one.