Exponential distribution

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.

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.

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.

The variance of X is given by

The exponential distribution with parameter λ is a continuous distribution whose probability density function is given by

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 quantile function (inverse cumulative distribution function) for Exp is

For example, the cumulative distribution function of Exponential (i.e.

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.

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.

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

The median of X is given by

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.

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.

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.

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.

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

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 ).

If a random variable X has this distribution, we write X ~ Exp.

which is the cumulative distribution function (CDF) of an exponential 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.

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

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

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

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

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.

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.

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