# 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

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.

### Dirac distribution

### Empirical distribution

### Laplace distribution

### Split-normal distribution