Cumulative Distribution Function

Definition (CDF)

Let $X\in \mathbb{R}$ be a random variable RV. The Cumulative Distribution Function (CDF) of $X$ denoted as $F_X$ or $F$ is a function

$$F_X:\mathbb{R}\to [0,1]$$

is defined by

$$F_X(x)=P(X\le x)$$

This function is

• non-decreasing
• $F_X(-\infty )=0$, an $F_X(\infty )=1$

Equivalently,

$$F_X(x)={\int }_{-\infty }^{x}f(x)dxx\in \mathbb{R}$$

the integral over the PDF.