Cumulative distribution Function

Definition Given $\mathbb{X}$, we define the cumulative distribution function (c.d.f) of $\mathbb{X}$ to be $F(x) = \mathbb{P}\{\mathbb{X} \leq x\}$

We denote $\mathbb{X}$ has c.d.f $F$ as $\mathbb{X} \sim F(x)$ for brevity. And note that $F$ is defined on $\mathbb{R}$.

Cumulative Distribution Function has some useful properties:

(a). c.d.f is right continuous.

Proof

From analysis, we know

$\lim_{n\to\infty}f(x_n)=f(x)\text{ for any sequence } \{x_n\} \text{ such that } x_n \to x$ if and only if $\lim_{t\to x}f(t)=f(x)$

And for any $\{x_n\}$ in $[x, \infty )$ such that $x_n\to x$, and let $A_n$ be $(-\infty, x_n]$

$F(\mathbb{X}\leq x) = \mathbb{P}( \mathbb{X} = \cap_{n=1}^\infty A_n ) \text{ (because } (-\infty, x] = \cap_{n=1}^\infty A_n )$ $= \mathbb{P}(\cap_{n=1}^\infty\{\mathbb{X} = A_n \}) = \cap_{n=1}^\infty\mathbb{P}(\mathbb{X} = A_n ) = \lim_{n\to\infty}\mathbb{P}(\mathbb{X} = A_n ) = \lim_{n\to\infty}\mathbb{P}(\mathbb{X} \leq x_n )$

The choice of $\{x_n\}$ was arbitrary, so

$\forall_x \lim_{t\to x^+}F(t) = F(x)$

(b). c.d.f is non-decreasing.

(c). $\lim_{t\to \infty}F(t) = 1, \lim_{t\to -\infty}F(t) = 0$

(d). If $a<b$, $\mathbb{P}( \mathbb{X}\leq b ) = \mathbb{P}( \mathbb{X} \leq a ) + \mathbb{P}( a < \mathbb{X} \leq b )$ (because $\{\mathbb{X}\leq a\}$ and $\{a<\mathbb{X}\leq b\}$ are disjoint set), so

$\mathbb{P}( a < \mathbb{X} \leq b ) = F(b) - F(a)$

C.d.f and its properties are often useful, because it is independent of the different types of random variable. These properties are true as long as it is a c.d.f of a random variable.