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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 ...
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.
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When the larger values tend to be farther away from the mean than the smaller values, one has a skew distribution to the right (i.e. there is positive skewness), one may for example select the log-normal distribution (i.e. the log values of the data are normally distributed), the log-logistic distribution (i.e. the log values of the data follow ...
In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable X is given by f X ( x ) = ∑ i = 1 n f Y i ( x ) p i , {\displaystyle f_{X}(x)=\sum _{i=1}^{n}f_{Y_{i}}(x)\;p_{i},}
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 ...
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The cross-entropy pendant is the expectation of the negative q-logarithm with respect to a second distribution, .So ().. Using =, this may be written ([]) /.For smaller , values all tend towards .