Simple continous distributions

Continous distributions

Exponential distribution

Weibull distribution

Power law

\(P(X)=\dfrac{\alpha -1}{a}(\dfrac{x}{a})^{-\alpha }\)

Where \(a\) is the lower bound.

\(P(X)=0\) for \(X<a\).

Moments of the power law

\(E[X^m]=\dfrac{\alpha - 1}{\alpha -1 -m }a\)

If \(m\ge \alpha -1 \) then this is not well defined.

Higher order moments, such that the variance, cannot be identified.

Logistic distribution

The logistic distribution has the cumulative distribution function:

\(F(x)=\dfrac{1}{1+e^{-\dfrac{x-\mu }{s}}} \)

Lévy distribution

Definition

The Lévy distribution is a continuous probability distribution.

The marginal probability is:

\(P(X)=\sqrt {\dfrac{c}{2\pi }}\dfrac{e^{-\dfrac{c}{2(x-\mu )}}}{(x-\mu )^{\dfrac{3}{2}}}\)

Other

Discrete uniform distribution

There is a set \(s\) such that:

\(P(x\in s)=p\)

\(P(x\not\in s)=0\)

Moments of the uniform distribution

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.

Dirac distribution

Empirical distribution

Laplace distribution

Split-normal distribution