There is a set \(s\) such that:
\(P(x\in s)=p\)
\(P(x\not\in s)=0\)
The mean is the mean of the set \(s\).
If the set is all numbers of the real line between two values, \(a\) and \(b\), then:
The mean is \(\dfrac{1}{2}(a+b)\).
The variance is \(\dfrac{(b-a)^2}{12}\) in the continuous case.
The variance is \(\dfrac{(b-a+1)^2-1}{12}\) in the discrete case.
\(P(X)=\dfrac{\alpha -1}{a}(\dfrac{x}{a})^{-\alpha }\)
Where \(a\) is the lower bound.
\(P(X)=0\) for \(X<a\).
\(E[X^m]=\dfrac{\alpha - 1}{\alpha -1 -m }a\)
If \(m\ge \alpha -1\) then this is not well defined.
Higher order moments, such that the variance, cannot be identified.
The logistic distribution has the cumulative distribution function:
\(F(x)=\dfrac{1}{1+e^{-\dfrac{x-\mu }{s}}}\)
The Lévy distribution is a continuous probability distribution.
The marginal probability is:
\(P(X)=\sqrt {\dfrac{c}{2\pi }}\dfrac{e^{-\dfrac{c}{2(x-\mu )}}}{(x-\mu )^{\dfrac{3}{2}}}\)