Stable distribution

Stable

Probability density function Symmetric α-stable distributions with unit scale factor Skewed centered stable distributions with unit scale factor
Cumulative distribution function CDFs for symmetric α-stable distributions CDFs for skewed centered stable distributions
Parameters	α ∈ (0, 2] — stability parameter β ∈ [−1, 1] — skewness parameter (note that skewness is undefined) c ∈ (0, ∞) — scale parameter μ ∈ (−∞, ∞) — location parameter
Support	x ∈ R, or x ∈ [μ, +∞) if α < 1 and β = 1, or x ∈ (-∞, μ] if α < 1 and β = −1
PDF	not analytically expressible, except for some parameter values
CDF	not analytically expressible, except for certain parameter values
Mean	μ when α > 1, otherwise undefined
Median	μ when β = 0, otherwise not analytically expressible
Mode	μ when β = 0, otherwise not analytically expressible
Variance	2c2 when α = 2, otherwise infinite
Skewness	0 when α = 2, otherwise undefined
Ex. kurtosis	0 when α = 2, otherwise undefined
Entropy	not analytically expressible, except for certain parameter values
MGF	undefined
CF	${\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},}$ where ${\displaystyle \Phi ={\begin{cases}\tan {\tfrac {\pi \alpha }{2}}&{\text{if }}\alpha \neq 1\\-{\tfrac {2}{\pi }}\log |t|&{\text{if }}\alpha =1\end{cases}}}$

α ∈ (0, 2] — stability parameter
β ∈ [−1, 1] — skewness parameter (note that skewness is undefined)
c ∈ (0, ∞) — scale parameter

${\displaystyle \exp \!{\Big [}\;it\mu -|c\,t|^{\alpha }\,(1-i\beta \operatorname {sgn}(t)\Phi )\;{\Big ]},}$

In probability theory, a distribution or a random variable is said to be stable if a linear combination of two independent copies of a random sample has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.