cumulative distribution function

Let $X$ be a random variable. Define $F_X:R\to [0,1]$ as$F_X(x)=\mathrm{Pr}[X\le x]$for all $x$. The function $F_X(x)$ is called the cumulative distribution function of $X$.

Every cumulative distribution function satisfies the following properties:

1. ${lim}_{x\to -\mathrm{\infty }}{F}_X(x)=0$ and ${lim}_{x\to +\mathrm{\infty }}{F}_X(x)=1$,

2. $F_X$ is a monotonically nondecreasing function,

3. $F_X$ is continuous from the right,

4. .

If $X$ is a discrete random variable, then the cumulative distribution can be expressed as$F_X(x)={\sum }_{k\le x}\mathrm{Pr}[X=k]$.

Similarly, if $X$ is a continuous random variable, then$F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(y)𝑑y$ where $f_X$ is the density distribution function.