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 ${F}_{X}(x)={\int}_{\mathrm{\infty}}^{x}{f}_{X}(y)\mathit{d}y$ where ${f}_{X}$ is the density distribution function.
Title  cumulative distribution function 
Canonical name  CumulativeDistributionFunction 
Date of creation  20130322 11:53:38 
Last modified on  20130322 11:53:38 
Owner  bbukh (348) 
Last modified by  bbukh (348) 
Numerical id  10 
Author  bbukh (348) 
Entry type  Definition 
Classification  msc 60A99 
Classification  msc 46L05 
Classification  msc 8200 
Classification  msc 8300 
Classification  msc 8100 
Related topic  DistributionFunction 
Related topic  DensityFunction 