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Random Variable Incidence Geometry Obvious Partial Orderings Of Subobjects Of An Object Function Subgraph Disjoint Measure Interval Borel Sigma Algebra Density Function One Sided Continuity

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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 is a distribution function if

is nondecreasing,
is continuous from the right,
$\lim_{x \rightarrow -\infty} F(x) = 0$ , 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 be a probability measure on $(\Omega, \mathcal{B})$ . Define by

$\begin{displaymath} F(x) = P((-\infty, x]). \end{displaymath}$

This particular

is called the distribution function of

. It is easy to verify that 1,2, and 3 hold for this

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 is a distribution function. We can define on a single half-open interval by

$\begin{displaymath} P((a,b]) = F(b) - F(a) \end{displaymath}$

and extend

to unions of disjoint intervals by

$\begin{displaymath} P( \cup_{i=1}^\infty (a_i, b_i])= \sum_{i=1}^\infty P((a_i, b_i]). \end{displaymath}$

and then further extend

to all the Borel subsets of $\mathbbmss{R}$ . It is clear that the distribution function of

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 on $\mathbbmss{R}$ defined as follows:

$\begin{displaymath} P_X(E) = P(X^{-1}(E)) \end{displaymath}$

for every Borel subset

of $\mathbbmss{R}$ .

is called the distribution of

. The distribution function of

$\begin{displaymath} F_X(x) = P(\omega \vert X(\omega) \leq x). \end{displaymath}$

The distribution function of

is also known as the law of

. Claim:

= the distribution function of

$\begin{eqnarray*} F_X(x) &=& P(\omega \vert X(\omega) \leq x) \\ &=& P(X^{-1}((-\infty, x]) \\ &=& P_X((-\infty, x]) \\ &=& F(x). \end{eqnarray*}$

Density Functions

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

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

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

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

Then it is clear that

satisfies the conditions 1,2,and 3 so

is a distribution function. The function

is called a density function for the distribution

If is a discrete random variable with density function and distribution function then

$\begin{displaymath}F(x)=\sum_{x_j\leq x} f(x_j).\end{displaymath}$

1	Partial Orderings Of Subobjects Of An Object
1	Random Variable
1	Incidence Geometry
1	Measure
1	Interval
1	Obvious
1	Disjoint
1	Function
1	Subgraph
3	Borel Sigma Algebra
4	Density Function
7	One Sided Continuity