Definition:Cumulative Distribution Function

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X$ be a random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

The cumulative distribution function (or c.d.f.) of $X$ is denoted $F \left({X}\right)$, and defined as:

$\forall x \in \R: F \left({X}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \Pr \left({X \le x}\right)$

It has the properties:

• $0 \le F \left({X}\right) \le 1$;

• $x_1 < x_2 \implies F \left({x_1}\right) \le F \left({x_2}\right)$;

• $\displaystyle\lim_{x \to -\infty} F \left({x}\right) = 0, \lim_{x \to \infty} F \left({x}\right) = 1$.

These are all demonstrated in Properties of Cumulative Distribution Function.

Notes

Some sources refer to this as a distribution function, but it can then become confused with the concept of a distribution function in physics.