distribution function
[this entry is currently being revised, so hold off on corrections until
this line is removed]
Let $F:\mathbb{R}\to \mathbb{R}$. Then $F$ is a distribution function^{} if

1.
$F$ is nondecreasing,

2.
$F$ is continuous from the right,

3.
${lim}_{x\to \mathrm{\infty}}F(x)=0$, and ${lim}_{x\to \mathrm{\infty}}F(x)=1$.
As an example, suppose that $\mathrm{\Omega}=\mathbb{R}$ and that $\mathcal{B}$ is the $\sigma $algebra of Borel subsets of $\mathbb{R}$. Let $P$ be a probability measure^{} on $(\mathrm{\Omega},\mathcal{B})$. Define $F$ by
$$F(x)=P((\mathrm{\infty},x]).$$ 
This particular $F$ is called the distribution function of $P$. It is easy to verify that 1,2, and 3 hold for this $F$.
In fact, every distribution function is the distribution function of some probability measure on the Borel subsets of $\mathbb{R}$. To see this, suppose that $F$ is a distribution function. We can define $P$ on a single halfopen interval by
$$P((a,b])=F(b)F(a)$$ 
and extend $P$ to unions of disjoint intervals by
$$P({\cup}_{i=1}^{\mathrm{\infty}}({a}_{i},{b}_{i}])=\sum _{i=1}^{\mathrm{\infty}}P(({a}_{i},{b}_{i}]).$$ 
and then further extend $P$ to all the Borel subsets of $\mathbb{R}$. It is clear that the distribution function of $P$ is $F$.
0.1 Random Variables
Suppose that $(\mathrm{\Omega},\mathcal{B},P)$ is a probability space and
$X:\mathrm{\Omega}\to \mathbb{R}$ is a random variable^{}. Then there is an
induced probability measure ${P}_{X}$ on $\mathbb{R}$ defined as
follows:
$${P}_{X}(E)=P({X}^{1}(E))$$ 
for every Borel subset $E$ of $\mathbb{R}$. ${P}_{X}$ is called the distribution^{} of $X$. The distribution function of $X$ is
$${F}_{X}(x)=P(\omega X(\omega )\le x).$$ 
The distribution function of $X$ is also known as the law of $X$. Claim: ${F}_{X}$ = the distribution function of ${P}_{X}$.
${F}_{X}(x)$  $=$  $P(\omega X(\omega )\le x)$  
$=$  $P({X}^{1}((\mathrm{\infty},x])$  
$=$  ${P}_{X}((\mathrm{\infty},x])$  
$=$  $F(x).$ 
0.2 Density Functions
Suppose that $f:\mathbb{R}\to \mathbb{R}$ is a nonnegative function such that
$${\int}_{\mathrm{\infty}}^{\mathrm{\infty}}f(t)\mathit{d}t=1.$$ 
Then one can define $F:\mathbb{R}\to \mathbb{R}$ by
$$F(x)={\int}_{\mathrm{\infty}}^{x}f(t)\mathit{d}t.$$ 
Then it is clear that $F$ satisfies the conditions 1,2,and 3 so $F$ is a distribution function. The function $f$ is called a density function for the distribution $F$.
If $X$ is a discrete random variable with density function $f$ and distribution function $F$ then
$$F(x)=\sum _{{x}_{j}\le x}f({x}_{j}).$$ 
Title  distribution function 
Canonical name  DistributionFunction 
Date of creation  20130322 13:02:51 
Last modified on  20130322 13:02:51 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  16 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 60E05 
Synonym  cumulative distribution function^{} 
Synonym  distribution 
Related topic  DensityFunction 
Related topic  CumulativeDistributionFunction 
Related topic  RandomVariable 
Related topic  Distribution 
Related topic  GeometricDistribution2 
Defines  law of a random variable 