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.,

$F(x)=\int _{-\infty }^{x}\,f(\lambda )\,d\lambda$

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

$f(x)={\frac {dF(x)}{dx}}$

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

(1) $F(x)\geq 0$ $\forall x$ inside its domain of support.

(2) $\lim _{x\to \infty }F(x)=1$, i.e., finitely convergent (to unity by convention).

(3) $F(a)\leq F(b)$ 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.