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| − | ''of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334801.png" />''
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| − | The function of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334802.png" /> taking at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334803.png" /> the value equal to the probability of the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334804.png" />.
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| − | Every distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334805.png" /> has the following properties:
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| − | 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334806.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334807.png" />;
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| − | 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334808.png" /> is left-continuous at every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334809.png" />;
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| − | 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348011.png" />. (Sometimes a distribution function is defined as the probability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348012.png" />; it is then right-continuous.)
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| − | In mathematical analysis, a distribution function is any function satisfying 1)–3). There is a one-to-one correspondence between the probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348013.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348014.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348015.png" /> of Borel subsets of the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348016.png" /> and the distribution functions. This correspondence is as follows: For any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348017.png" />,
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348018.png" /></td> </tr></table>
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| − | Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348019.png" /> satisfying 1)–3) can be regarded as the distribution function of some random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348020.png" /> (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348021.png" />) defined on the probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348022.png" />.
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| − | Any distribution function can be uniquely written as a sum
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348023.png" /></td> </tr></table>
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348024.png" /> are non-negative numbers with sum equal to 1, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348025.png" /> are distribution functions such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348026.png" /> is absolutely continuous,
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348027.png" /></td> </tr></table>
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348028.png" /> is a "step-function" ,
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348029.png" /></td> </tr></table>
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| − | where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348030.png" /> are the points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348031.png" /> "jumps" and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348032.png" /> are proportional to the size of these jumps, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348033.png" /> is the "singular" component — a continuous function whose derivative is zero almost-everywhere.
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| − | Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348035.png" /> be an infinite sequence of independent random variables assuming the values 1 and 0 with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348037.png" />, respectively. Also, let
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348038.png" /></td> </tr></table>
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| − | Now:
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| − | 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348039.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348041.png" /> has an absolutely-continuous distribution function (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348043.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348044.png" /> is uniformly distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348045.png" />);
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| − | 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348047.png" /> has a "step" distribution function (it has jumps at all the dyadic-rational points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348048.png" />);
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| − | 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348050.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348052.png" /> has a "singular" distribution function.
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| − | 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.
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| − | The "distance" between two distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348054.png" /> on the real line is often defined in terms of the corresponding distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348056.png" />, by putting, for example,
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348057.png" /></td> </tr></table>
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| − | or
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348058.png" /></td> </tr></table>
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| − | (see [[Distributions, convergence of|Distributions, convergence of]]; [[Lévy metric|Lévy metric]]; [[Characteristic function|Characteristic function]]).
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| − | The distribution functions of the probability distributions most often used (e.g. the normal, binomial and Poisson distributions) have been tabulated.
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| − | To test hypotheses concerning a distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348059.png" /> using results of independent observations, one can use some measure of the deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348060.png" /> from the empirical distribution function (see [[Kolmogorov test|Kolmogorov test]]; [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov test]]; [[Cramér–von Mises test|Cramér–von Mises test]]).
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| − | 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.
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| − | For a more detailed treatment of distribution functions see [[Gram–Charlier series|Gram–Charlier series]]; [[Edgeworth series|Edgeworth series]]; [[Limit theorems|Limit theorems]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Random variables and probability distributions" , Cambridge Univ. Press (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1957–1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table>
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| − | ====Comments====
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| − | In the Russian literature distributions functions are taken to be left-continuous. In the Western literature it is common to define them to be right-continuous. Thus, the distribution function of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348061.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348062.png" />. It then has the properties 1); 2') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348063.png" /> is right-continuous at every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348064.png" />; 3). The unique probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348065.png" /> corresponding to it is now defined as
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348066.png" /></td> </tr></table>
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| − | while the "step-function" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348067.png" /> in the above-mentioned decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348068.png" /> is
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348069.png" /></td> </tr></table>
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.L. Johnson, S. Kotz, "Distributions in statistics" , Houghton Mifflin (1970)</TD></TR></table>
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