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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 ...
Cumulative distribution function for the exponential distribution, often used as the cumulative failure function ().. To accurately model failures over time, a cumulative failure distribution, () must be defined, which can be any cumulative distribution function (CDF) that gradually increases from to .
The logistic distribution is a special case of the Tukey lambda distribution. Specification. Cumulative distribution function ... If X ~ Exponential(1) then
The Erlang distribution is the distribution of the sum of k independent and identically distributed random variables, each having an exponential distribution. The long-run rate at which events occur is the reciprocal of the expectation of X , {\displaystyle X,} that is, λ / k . {\displaystyle \lambda /k.}
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a / base measure) for a random variable X for which E[X] = αθ = α/λ is fixed and greater than zero, and E[ln X] = ψ(α) + ln θ = ψ(α) − ln λ is fixed (ψ is the digamma function).
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ 2, and Y is ...
The exponential distribution is the continuous analogue of the geometric distribution. Applying the floor function to the exponential distribution with parameter λ {\displaystyle \lambda } creates a geometric distribution with parameter p = 1 − e − λ {\displaystyle p=1-e^{-\lambda }} defined over N 0 {\displaystyle \mathbb {N} _{0}} .
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 ...