# Distribution function

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of a random variable The function of a real variable taking at each the value equal to the probability of the inequality .

Every distribution function has the following properties:

1) when ;

2) is left-continuous at every ;

3) , . (Sometimes a distribution function is defined as the probability of ; it is then right-continuous.)

In mathematical analysis, a distribution function is any function satisfying 1)–3). There is a one-to-one correspondence between the probability distributions on the -algebra of Borel subsets of the real line and the distribution functions. This correspondence is as follows: For any interval , Any function satisfying 1)–3) can be regarded as the distribution function of some random variable (e.g. ) defined on the probability space .

Any distribution function can be uniquely written as a sum where are non-negative numbers with sum equal to 1, and are distribution functions such that is absolutely continuous,  is a "step-function" , where the are the points where "jumps" and the are proportional to the size of these jumps, and is the "singular" component — a continuous function whose derivative is zero almost-everywhere.

Example. Let , be an infinite sequence of independent random variables assuming the values 1 and 0 with probabilities and , respectively. Also, let Now:

1) if for all , then has an absolutely-continuous distribution function (with for , that is, is uniformly distributed on );

2) if , then has a "step" distribution function (it has jumps at all the dyadic-rational points in );

3) if and as , then has a "singular" distribution function.

This example serves to illustrate the theorem of P. Lévy asserting that the limit of an infinite convolution of discrete distribution functions can contain only one of the components mentioned above.

The "distance" between two distributions and on the real line is often defined in terms of the corresponding distribution functions and , by putting, for example, or The distribution functions of the probability distributions most often used (e.g. the normal, binomial and Poisson distributions) have been tabulated.

To test hypotheses concerning a distribution function using results of independent observations, one can use some measure of the deviation of from the empirical distribution function (see Kolmogorov test; Kolmogorov–Smirnov test; Cramér–von Mises test).

The concept of a distribution function can be extended in a natural way to the multi-dimensional case, but multi-dimensional distribution functions are significantly less used in comparison to one-dimensional distribution functions.

For a more detailed treatment of distribution functions see Gram–Charlier series; Edgeworth series; Limit theorems.

How to Cite This Entry:
Distribution function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_function&oldid=17094
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article