# 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).