# cumulative distribution function

Let $X$ be a random variable. Define $F_{X}\colon R\to[0,1]$ as $F_{X}(x)=\operatorname{Pr}[X\leq x]$ 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. 1.

$\lim_{x\to-\infty}{F_{X}(x)}=0$ and $\lim_{x\to+\infty}{F_{X}(x)}=1$,

2. 2.

$F_{X}$ is a monotonically nondecreasing function,

3. 3.

$F_{X}$ is continuous from the right,

4. 4.

$\operatorname{Pr}[a.

If $X$ is a discrete random variable, then the cumulative distribution can be expressed as $F_{X}(x)=\sum_{k\leq x}\operatorname{Pr}[X=k]$.

Similarly, if $X$ is a continuous random variable, then $F_{X}(x)=\int_{-\infty}^{x}f_{X}(y)dy$ 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