Distribution Function

Definition

$F_X(x):\mathbb{R}\to [0,1]:={\mu }_X((-\infty ,x])=P(X\le x)$

A distribution function $F_X$ is a function for the Random Variable on Probability Space $(\Omega ,\mathcal{F},P)$

Facts

$(\Omega ,\mathcal{F},P)\sim (\mathbb{R},\mathcal{R},{\mu }_X)$

Proof By definition, $F_X(x)={\mu }_X((-\infty ,x])$ So, defining $\forall x\in \mathbb{R},F_X(x)$ is the Equivalence Relation to defining $\mathcal{A}=\{(-\infty ,x]:x\in \mathbb{R}\},{\mu }_X:\mathcal{A}\to [0,1]$

Since $\mathcal{A}$ is a Pi-System, ${\mu }_X:\sigma (\mathcal{R})=\mathcal{R}\to [0,1]$ is uniquely determined by ${\mu }_X:\mathcal{A}\to [0,1]$ by Extension from Pi-System

Now, $(\mathbb{R},\mathcal{R},{\mu }_X)$ is a Measurable Space induced by $X$. Therefore, defining ${\mu }_X$ on $(\mathbb{R},\mathcal{R})$ is equivalent to the defining $P$ on $(\Omega ,\mathcal{F})$

Distribution function(CDF) $F:\mathbb{R}\to [0,1]$ has the following properties

• Monotonic increasing: $\forall a,b,\in \mathbb{R},a<b\Rightarrow F(a)\le F(b)$
• $\underset{x\to -\infty }{\lim }F(x)=0$
• $\underset{x\to \infty }{\lim }F(x)=1$
• Right-continuous: $\underset{x\to x_0^{+}}{\lim }F(x)=F(x_0)$

If a function $F:\mathbb{R}\to [0,1]$ satisfies the following properties, then $F$ is a distribution function(CDF) of some Random Variable $X$

• Monotonic increasing: $\forall a,b,\in \mathbb{R},a<b\Rightarrow F(a)\le F(b)$
• $\underset{x\to -\infty }{\lim }F(x)=0$
• $\underset{x\to \infty }{\lim }F(x)=1$
• Right-continuous: $\underset{x\to x_0^{+}}{\lim }F(x)=F(x_0)$

$\forall a,b\in \mathbb{R},a<b\Rightarrow (P(a<X\le b)=F_X(b)-F_X(a))$