# Cumulative distribution function

In probability theory, a **cumulative distribution function** *F(x)* of a probability density function say *f(x)* is a real valued and continuous function whose value is the proportion of probability values of a variable which occur on the part of the real line up and including the value of that variable; i.e.,

Considering this definition in light of the Fundamental Theorem of Calculus yields:

Due to the properties of the probability density function *f(x)*, the cumulative distribution function *F(x)* will have the following properties:

(1) inside its domain of support.

(2) , i.e., finitely convergent (to unity by convention).

(3) for a<b, i.e., is non-decreasing

If the domain of the variable is finite, then the upper limit in equation (2) above should be the upper bound of the variables domain of support.