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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 ...
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The Dagum distribution; The exponential distribution, which describes the time between consecutive rare random events in a process with no memory. The exponential-logarithmic distribution; The F-distribution, which is the distribution of the ratio of two (normalized) chi-squared-distributed random variables, used in the analysis of variance.
For example, the Bernoulli distribution is a binomial distribution with n = 1 trial, the exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ), and the geometric distribution is a special case of the negative binomial distribution. Some exponential family distributions are not NEF. The lognormal and Beta ...
In probability theory, 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 ...
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 ...
The probability density function of the wrapped exponential distribution is [1] (;) = = (+) =,for < where > is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range <.
The q-deformed exponential and logarithmic functions were first introduced in Tsallis statistics in 1994. [1] However, the q -logarithm is the Box–Cox transformation for q = 1 − λ {\displaystyle q=1-\lambda } , proposed by George Box and David Cox in 1964.