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Homedistribution function

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# distribution function

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

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

## Mathematics Subject Classification

60E05*no label found*

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## Attached Articles

## Corrections

reword by Mathprof ✓

order of definition by CWoo ✓

synonym by CWoo ✓

add synonym to "also defines" box by CWoo ✓