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 $F_X$ be the cumulative distribution function.

That is:

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

for each $x \in \R$.

Then $F_X$ has the following properties:

Bounds for Cumulative Distribution Function

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

Cumulative Distribution Function is Increasing

$F_X$ is an increasing function.

Cumulative Distribution Function is Right-Continuous

$F_X$ is right-continuous.

Limit of Cumulative Distribution Function at Positive Infinity

$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$

where $\ds \lim_{x \mathop \to \infty} \map {F_X} x$ denotes the limit at $+\infty$ of $F_X$.

Limit of Cumulative Distribution Function at Negative Infinity

$\ds \lim_{x \mathop \to -\infty} \map {F_X} x = 0$

where $\ds \lim_{x \mathop \to -\infty} \map {F_X} x$ denotes the limit at $-\infty$ of $F_X$.