# Simple continous distributions

## Continous distributions

### 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.