Properties of Cumulative Distribution Function

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\map F X$ be the cumulative distribution function of $X$:
 * $\forall x \in \R: \map F x = \map \Pr {X \le x}$

Then the following conditions apply to $\map F X$:

Bounds of CDF

 * $0 \le \map F X \le 1$

CDF is Increasing

 * $x_1 < x_2 \implies \map F {x_1} \le \map F {x_2}$

That is, $F$ is an increasing mapping.

Limits of CDF

 * $\displaystyle \lim_{x \mathop \to -\infty} \map F x = 0$
 * $\displaystyle \lim_{x \mathop \to \infty} \map F x = 1$

Proof of Bounds of CDF
This follows directly from the definition of $\Pr$.

Proof that CDF is Increasing
Let $x, y \in \R: x \le y$.

Let $\map X \omega \le x$.

Then:
 * $\map X \omega \le y$

and so:
 * $\set {\omega \in \Omega: \map X \omega \le x} \subseteq \set {\omega \in \Omega: \map X \omega \le y}$

Hence the result.

Proof of Limits of CDF
As $x \to -\infty$, $\hointl \gets x \to \O$.

So:
 * $\map {X^{-1} } {\hointl \gets x} \to \O$

and so:
 * $\map F x \to 0$

Similarly, as $x \to +\infty$, $\hointl \gets x \to \R$.

So:
 * $\map {X^{-1} } {\hointl \gets x} \to \Omega$

and so:
 * $\map F x \to 1$