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distribution function (Definition)

[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. $F$ 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

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

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

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

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

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:

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

for every Borel subset $E$ of $\mathbbmss{R}$. $P_X$ is called the distribution of $X$. The distribution function of $X$ is
\begin{displaymath} F_X(x) = P(\omega \vert X(\omega) \leq x). \end{displaymath}

The distribution function of $X$ is also known as the law of $X$. Claim: $F_X$ = the distribution function of $P_X$.
\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 $F$ 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


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



"distribution function" is owned by Mathprof. [ full author list (2) | owner history (2) ]
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See Also: density function, cumulative distribution function, random variable, probability distribution function, geometric distribution

Other names:  cumulative distribution function, distribution
Also defines:  law of a random variable

Attachments:
type of a distribution function (Definition) by CWoo
multivariate distribution function (Definition) by CWoo
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Cross-references: discrete random variable, density function, function, induced, random variable, probability space, clear, intervals, disjoint, unions, half-open interval, probability measure, Borel subsets, continuous from the right, line
There are 51 references to this entry.

This is version 13 of distribution function, born on 2002-09-11, modified 2006-11-20.
Object id is 3451, canonical name is DistributionFunction.
Accessed 21442 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
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