Poisson distribution
Weibull and Stable count

Poisson's probability mass function f (k; λ) {\displaystyle f(k;\lambda)} can be expressed in a form similar to the product distribution of a Weibull distribution and a variant form of the stable count distribution. The variable (k + 1) {\displaystyle (k+1)} can be regarded as inverse of Lévy's stability parameter in the stable count distribution: f (k; λ) = ∫ 0 ∞ 1 u W k + 1 (λ u) d u, {\displaystyle f(k;\lambda)=\displaystyle \int _{0}^{\infty }{\frac {1}{u}}\,W_{k+1}({\frac {\lambda }{u}})\left\,du,} where N α (ν) {\displaystyle {\mathfrak {N}}_{\alpha }(\nu)} is a standard stable count distribution of shape α = 1 / (k + 1), {\displaystyle \alpha =1/\left(k+1\right),} and W k + 1 (x) {\displaystyle W_{k+1}(x)} is a standard Weibull distribution of shape k + 1. {\displaystyle k+1.}
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