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:


 * $$0 \le F \left({X}\right) \le 1$$;


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


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

These are all demonstrated in Properties of Cumulative Distribution Function.