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In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace.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 along the abscissa, although the term is also sometimes used to refer to ...
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
Laplace distribution, or bilateral exponential distribution, consisting of two exponential distributions glued together on each side of a threshold; Gumbel distribution, the cumulative distribution function of which is an iterated exponential function (the exponential of an exponential function).
The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. If X is a Student’s t random variable with ν degree of freedom, then X 2 is an F (1,ν) random variable. If X is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ.
This can also be seen as a three-parameter generalization of a normal distribution to add skew; another distribution like that is the skew normal distribution, which has thinner tails. The distribution is a compound probability distribution in which the mean of a normal distribution varies randomly as a shifted exponential distribution.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
They may also be generated as the difference of two exponential distributions. If X 1 is drawn from exponential distribution with mean and rate (m 1,λ/κ) and X 2 is drawn from an exponential distribution with mean and rate (m 2,λκ) then X 1 - X 2 is distributed according to the asymmetric Laplace distribution with parameters (m1-m2, λ, κ)
The Erlang distribution has two parameters, the shape an integer k > 0 and the rate λ > 0. This is sometimes denoted E ( k ,λ). The Erlang distribution can be written in the form of a phase-type distribution by making S a k × k matrix with diagonal elements -λ and super-diagonal elements λ, with the probability of starting in state 1 equal ...