# distribution function

[this entry is currently being revised, so hold off on corrections until this line is removed]

Let $F:\mathbbmss{R}\to\mathbbmss{R}$. Then $F$ is a distribution function if

1. 1.

$F$ is nondecreasing,

2. 2.

$F$ is continuous from the right,

3. 3.

$\lim_{x\rightarrow-\infty}F(x)=0$, and $\lim_{x\rightarrow\infty}F(x)=1$.

As an example, suppose that $\Omega=\mathbbmss{R}$ and that $\mathcal{B}$ is the $\sigma$-algebra of Borel subsets of $\mathbbmss{R}$. Let $P$ be a probability measure on $(\Omega,\mathcal{B})$. Define $F$ by

 $F(x)=P((-\infty,x]).$

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 $\mathbbmss{R}$. To see this, suppose that $F$ is a distribution function. We can define $P$ on a single half-open interval by

 $P((a,b])=F(b)-F(a)$

and extend $P$ to unions of disjoint intervals by

 $P(\cup_{i=1}^{\infty}(a_{i},b_{i}])=\sum_{i=1}^{\infty}P((a_{i},b_{i}]).$

and then further extend $P$ to all the Borel subsets of $\mathbbmss{R}$. It is clear that the distribution function of $P$ is $F$.

## 0.1 Random Variables

Suppose that $(\Omega,\mathcal{B},P)$ is a probability space and $X:\Omega\to\mathbbmss{R}$ is a random variable. Then there is an induced probability measure $P_{X}$ on $\mathbbmss{R}$ defined as follows:

 $P_{X}(E)=P(X^{-1}(E))$

for every Borel subset $E$ of $\mathbbmss{R}$. $P_{X}$ is called the distribution of $X$. The distribution function of $X$ is

 $F_{X}(x)=P(\omega|X(\omega)\leq x).$

The distribution function of $X$ is also known as the law of $X$. Claim: $F_{X}$ = the distribution function of $P_{X}$.

 $\displaystyle F_{X}(x)$ $\displaystyle=$ $\displaystyle P(\omega|X(\omega)\leq x)$ $\displaystyle=$ $\displaystyle P(X^{-1}((-\infty,x])$ $\displaystyle=$ $\displaystyle P_{X}((-\infty,x])$ $\displaystyle=$ $\displaystyle F(x).$

## 0.2 Density Functions

Suppose that $f:\mathbbmss{R}\to\mathbbmss{R}$ is a nonnegative function such that

 $\int_{-\infty}^{\infty}f(t)dt=1.$

Then one can define $F:\mathbbmss{R}\to\mathbbmss{R}$ by

 $F(x)=\int_{-\infty}^{x}f(t)dt.$

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

 $F(x)=\sum_{x_{j}\leq x}f(x_{j}).$
 Title distribution function Canonical name DistributionFunction Date of creation 2013-03-22 13:02:51 Last modified on 2013-03-22 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