distribution function

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

1.
$F$ is nondecreasing,
2.
$F$ is continuous from the right,
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 iseasy to verify that 1,2, and 3 hold for this $F$ .

In fact, every distribution function is the distribution function of someprobability 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-openinterval 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 aninduced probability measure $P_{X}$ on $\\mathbbmss{R}$ defined asfollows:

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

for every Borel subset $E$ of $\\mathbbmss{R}$ . $P_{X}$ is called thedistribution of $X$ . The distribution functionof $X$ 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 functionsuch 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 functionfor the distribution $F$ .

If $X$ is a discrete random variable with density function $f$ and distributionfunction $F$ then

F(x)=\\sum_{x_{j}\\leq x}f(x_{j}).