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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 ...
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 ...
The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include: mounded (or unimodal), U-shaped, J-shaped, reverse-J shaped and multi-modal. [1] A bimodal distribution would have two high points rather than one. The shape of a distribution is ...
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 terms "distribution" and "family" are often used loosely: Specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; [a] however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families ...
Exponential distribution and memorylessness proof [ edit ] The only memoryless continuous probability distribution is the exponential distribution, shown in the following proof: [ 9 ]
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 ...
Using the standard formalism of probability theory, let and be two random variables defined on probability spaces (,,) and (,,).Then a coupling of and is a new probability space (,,) over which there are two random variables and such that has the same distribution as while has the same distribution as .