Cumulative Distribution Function

A function that determine the probability that a RV $X$ will take a value less than or equal to $x$. Oftentimes messy to compute.

Notations

• $F_X(x)\equiv P(X\le x)=P(\{s\in S|X(s)\le x\}),\forall x$ represents probability of random variable being less or equal to value $x$
• $F_X(x)\equiv P(a<X\le b)=P(X\le b)-P(X\le a)=F_X(b)-F_X(a)$ represents probability of random $X$ being in any interval $(a,b]$

Axioms

• $F_X(\infty )={\lim }_{x\to -\infty }{F}_X(x)=1$
• $F_X(-\infty )={\lim }_{x\to \infty }{F}_X(x)=0$
• $\forall x_1<x_2\in \mathbb{R}⟹F_X(x_1)\le F_X(x_2)$
• $F_X(\cdot )$ is right-continuous