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Related to this distribution are a number of other distributions: the displaced Poisson, the hyper-Poisson, the general Poisson binomial and the Poisson type distributions. The Conway–Maxwell–Poisson distribution, a two-parameter extension of the Poisson distribution with an adjustable rate of decay.
However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution. Both the hyperbolic secant distribution and the logistic distribution are special cases of the Champernowne distribution, which has exponential tails.
In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve). Main article: Central limit theorem The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution.
The reciprocal 1/X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples: If X is a Cauchy (μ, σ) random variable, then 1/X is a Cauchy (μ/C, σ/C) random variable where C = μ 2 + σ 2.
The mutual information of two multivariate normal distribution is a special case of the Kullback–Leibler divergence in which is the full dimensional multivariate distribution and is the product of the and dimensional marginal distributions and , such that + =.
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y , the distribution of the random variable Z that is formed as the product Z = X Y {\displaystyle Z=XY} is a product distribution .
The formulation discussed above originates from John. [3] The literature offers two mathematically equivalent alternative parameterizations . Britton, Fisher and Whitley [8] offer a parameterization if terms of mode, dispersion and normed skewness, denoted with (,,).
A singular distribution, generalised distribution, or Stefan-Sussmann distribution, is a smooth distribution which is not regular. This means that the subspaces Δ x ⊂ T x M {\displaystyle \Delta _{x}\subset T_{x}M} may have different dimensions, and therefore the subset Δ ⊂ T M {\displaystyle \Delta \subset TM} is no longer a smooth ...