Bounds for 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 $F_X$ be the cumulative distribution function.

Then:

$0 \le \map {F_X} x \le 1$ for each $x \in \R$

Proof

From the definition of the cumulative distribution function, we have:

$\map {F_X} x = \map \Pr {X \le x}$

for each $x \in \R$.

We have:

$\O \subseteq \set {\omega \in \Omega : \map X \omega \le x} \subseteq \Omega$

So, from Measure is Monotone, we have:

$\map \Pr \O \le \map \Pr {X \le x} \le \map \Pr \Omega$

From the definition of a probability measure, we have:

$\map \Pr \O = 0$

and:

$\map \Pr \Omega = 1$

so:

$0 \le \map \Pr {X \le x} \le 1$

for each $x \in \R$.

$\blacksquare$