Properties of Cumulative Distribution Function

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

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

Let $$F \left({X}\right)$$ be the cumulative distribution function of $$X$$, that is:
 * $$\forall x \in \R: F \left({X}\right) = \Pr \left({X \le x}\right)$$

Then the following conditions apply to $$F \left({X}\right)$$:

Bounds of CDF

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

CDF is Increasing

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

That is, $$F$$ is an increasing mapping.

Limits of CDF

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

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

$$ $$ $$

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

Let $$X \left({\omega}\right) \le x$$.

Then $$X \left({\omega}\right) \le y$$, and so:
 * $$\left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \subseteq \left\{{\omega \in \Omega: X \left({\omega}\right) \le y}\right\}$$

Hence the result.

Proof of Limits of CDF
As $$x \to -\infty$$, $$\left({-\infty \, . \, . \, x}\right] \to \varnothing$$.

So $$X^{-1} \left({\left({-\infty \,. \, . \, x}\right]}\right) \to \varnothing$$ and so $$F \left({x}\right) \to 0$$.

Similarly, as $$x \to +\infty$$, $$\left({-\infty \, . \, . \, x}\right] \to \R$$.

So $$X^{-1} \left({\left({-\infty \,. \, . \, x}\right]}\right) \to \Omega$$ and so $$F \left({x}\right) \to 1$$.