Cumulative distribution function

A 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.,

<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>

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

<math> f(x) = \frac{dF(x)}{dx}</math>

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

(1) <math> F(x) \geq 0 </math> <math> \forall x </math> inside its domain of support.

(2) <math> \lim_{x \to \infty}F(x) = 1 </math>, i.e., finitely convergent (to unity by convention).

(3) <math> F(a) \leq F(b) </math> 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.