The (cumulative) distribution function, or *probability distribution function*, *F*(*x*) is the mathematical equation that describes the probability that a variable *X* is less that or equal to *x*, i.e.

*F*(*x*) = *P*(*X*≤*x*) for all *x*

where *P*(*X*≤*x*) means the probability of the event *X*≤*x*.

A cumulative distribution function has the following properties:

1. *F(x)* is always non-decreasing, i.e. \frac{d}{dx} F(x)\geq 0

2. *F*(*x*) = 0 at *x* = -∞ or minimum

* F*(*x*) = 1 at *x* = ∞ or maximum