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.

Title	cumulative distribution function
Canonical name	CumulativeDistributionFunction
Date of creation	2013-03-22 11:53:38
Last modified on	2013-03-22 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 82-00
Classification	msc 83-00
Classification	msc 81-00
Related topic	DistributionFunction
Related topic	DensityFunction