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''of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334801.png" />''
+
''of a random variable $X$''
  
 
{{MSC|60E05}}
 
{{MSC|60E05}}
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[[Category:Distribution theory]]
 
[[Category:Distribution theory]]
  
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" />.
+
The function of a real variable $x$ taking at each $x$ the value equal to the probability of the inequality $X < x$.
  
Every distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d0334805.png" /> has the following properties:
+
Every distribution function $F(x)$ has the following properties:
  
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" />;
+
1) $F(x') \le F(x'')$ when $x' < x''$;
  
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" />;
+
2) $F(x)$ is left-continuous at every $x$;
  
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.)
+
3) $\lim\limits_{x \rightarrow -\infty} F(x) = 0$, $\lim\limits_{x \rightarrow +\infty} F(x) = 1$. (Sometimes a distribution function is defined as the probability of $X \le x$; 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 <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" />,
+
In mathematical analysis, a distribution function is any function satisfying 1)–3). There is a one-to-one correspondence between the probability distributions $P_{F}$ on the $\sigma$-algebra $\mathcal{B}$ of Borel subsets of the real line $\mathbb{R}^{1}$ and the distribution functions. This correspondence is as follows: For any interval $\left[ a, b \right]$,
  
<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>
+
$$
 +
P_{F}(a, b) = F(b) - F(a)
 +
$$
  
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" />.
+
 
 +
Any function $F$ satisfying 1)–3) can be regarded as the distribution function of some random variable $X$ (e.g. $X(x) = x$) defined on the probability space $\left( \mathbb{R}^1, \mathcal{B}, P_{F} \right)$.
  
 
Any distribution function can be uniquely written as a sum
 
Any distribution function can be uniquely written as a sum
  
<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>
+
$$
 +
F(x) = \alpha_{1} F_{1}(x) + \alpha_{2} F_{2}(x) + \alpha_{3} F_{3}(x),
 +
$$
 +
 
 +
where $\alpha_{1}, \alpha_{2}, \alpha_{3}$ are non-negative numbers with sum equal to 1, and $F_{1}, F_{2}, F_{3}$ are distribution functions such that $F_{1}(x)$ is absolutely continuous,
  
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,
+
$$
 +
F_{1}(x) = \int\limits_{-\infty}^{x} p(z) dz,
 +
$$
  
<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>
+
$F_{2}(x)$ is a "step-function",
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033480/d03348028.png" /> is a "step-function" ,
+
$$
 +
F_{2}(x) = \sum\limits_{x_{k} < x} p_{k},
 +
$$
  
<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>
+
where the $x_{k}$ are the points where $F(x)$ "jumps" and the $p_{k} > 0$ are proportional to the size of these jumps, and $F_{3}(x)$ is the "singular" component — a continuous function whose derivative is zero almost-everywhere.
  
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.
 
  
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
+
Example. Let $X_{k}$, $k = 1, 2, \ldots,$ be an infinite sequence of independent random variables assuming the values 1 and 0 with probabilities $0 < p_{k} \le \frac{1}{2}$ and $q_{k} = 1 - p_{k}$, respectively. Also, let
  
<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>
+
$$
 +
X = \sum\limits_{k = 1}^{\infty} \frac{X_{k}}{2^{k}}
 +
$$
  
 
Now:
 
Now:
  
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" />);
+
1) if $p_k = q_k = \frac{1}{2}$ for all $k$, then $X$ has an absolutely-continuous distribution function (with $p(x) = 1$ for $0 \le x \le 1$, that is, $X$ is uniformly distributed on $\left[ 0, 1 \right]$);
  
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" />);
+
2) if $\sum\limits_{k = 1}^{\infty} p_k < \infty$, then $X$ has a "step" distribution function (it has jumps at all the dyadic-rational points in $\left[ 0, 1 \right]$);
  
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.
+
3) if $\sum\limits_{k = 1}^{\infty} p_k = \infty$ and $p_k \rightarrow 0$ as $k \rightarrow \infty$, then $X$ 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.
 
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 <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,
+
The "distance" between two distributions $P$ and $Q$ on the real line is often defined in terms of the corresponding distribution functions $F$ and $S$, by putting, for example,
  
<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>
+
$$
 +
\rho_1(P, Q) = \sup_{x} \left| F(x) - S(x) \right|
 +
$$
  
 
or
 
or
  
<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>
+
$$
 +
\rho_2(P, Q) = \mathrm{Var} \left( F(x) - S(x) \right)
 +
$$
  
 
(see [[Distributions, convergence of|Distributions, convergence of]]; [[Lévy metric|Lévy metric]]; [[Characteristic function|Characteristic function]]).
 
(see [[Distributions, convergence of|Distributions, convergence of]]; [[Lévy metric|Lévy metric]]; [[Characteristic function|Characteristic function]]).
Line 61: Line 77:
 
The distribution functions of the probability distributions most often used (e.g. the normal, binomial and Poisson distributions) have been tabulated.
 
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 <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]]).
+
To test hypotheses concerning a distribution function $F$ using results of independent observations, one can use some measure of the deviation of $F$ from the empirical distribution function (see [[Kolmogorov test|Kolmogorov test]]; [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov test]]; [[Cramér–von Mises 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.
 
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.
Line 68: Line 84:
  
 
====References====
 
====References====
 +
 
{|
 
{|
 +
 
|valign="top"|{{Ref|C}}|| H. Cramér, "Random variables and probability distributions", Cambridge Univ. Press (1970) {{MR|0254895}} {{ZBL|0184.40101}}
 
|valign="top"|{{Ref|C}}|| H. Cramér, "Random variables and probability distributions", Cambridge Univ. Press (1970) {{MR|0254895}} {{ZBL|0184.40101}}
 +
 
|-
 
|-
 +
 
|valign="top"|{{Ref|C2}}|| H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
 
|valign="top"|{{Ref|C2}}|| H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
 +
 
|-
 
|-
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1957–1971)
+
 
 +
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1957–1971)
 +
 
 
|-
 
|-
 +
 
|valign="top"|{{Ref|BS}}|| 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) {{MR|}} {{ZBL|0529.62099}}
 
|valign="top"|{{Ref|BS}}|| 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) {{MR|}} {{ZBL|0529.62099}}
 +
 
|}
 
|}
  
 
====Comments====
 
====Comments====
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
 
  
<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>
+
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 $X$ is the function $F(x) = \mathrm{P} \lbrace X \le x \rbrace$. It then has the properties 1); 2') $F(x)$ is right-continuous at every $x$; 3). The unique probability distribution $P_{F}$ corresponding to it is now defined as
  
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
+
$$
 +
P_{F}(a, b) = F(b) - F(a)
 +
$$
  
<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>
+
while the "step-function" $F_{2}(x)$ in the above-mentioned decomposition $F = \alpha_{1} F_{1} + \alpha_{2} F_{2} + \alpha_{3} F_{3} $ is
 +
 
 +
$$
 +
F_{2} (x) = \sum\limits_{x_{k} \le x} p_{k}.
 +
$$
  
 
====References====
 
====References====
 +
 
{|
 
{|
 +
 
|valign="top"|{{Ref|JK}}|| N.L. Johnson, S. Kotz, "Distributions in statistics" , Houghton Mifflin (1970) {{MR|0270476}} {{MR|0270475}} {{ZBL|0213.21101}}
 
|valign="top"|{{Ref|JK}}|| N.L. Johnson, S. Kotz, "Distributions in statistics" , Houghton Mifflin (1970) {{MR|0270476}} {{MR|0270475}} {{ZBL|0213.21101}}
 +
 
|}
 
|}

Revision as of 18:56, 29 July 2013

of a random variable $X$

2010 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

The function of a real variable $x$ taking at each $x$ the value equal to the probability of the inequality $X < x$.

Every distribution function $F(x)$ has the following properties:

1) $F(x') \le F(x'')$ when $x' < x''$;

2) $F(x)$ is left-continuous at every $x$;

3) $\lim\limits_{x \rightarrow -\infty} F(x) = 0$, $\lim\limits_{x \rightarrow +\infty} F(x) = 1$. (Sometimes a distribution function is defined as the probability of $X \le x$; 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 $P_{F}$ on the $\sigma$-algebra $\mathcal{B}$ of Borel subsets of the real line $\mathbb{R}^{1}$ and the distribution functions. This correspondence is as follows: For any interval $\left[ a, b \right]$,

$$ P_{F}(a, b) = F(b) - F(a) $$


Any function $F$ satisfying 1)–3) can be regarded as the distribution function of some random variable $X$ (e.g. $X(x) = x$) defined on the probability space $\left( \mathbb{R}^1, \mathcal{B}, P_{F} \right)$.

Any distribution function can be uniquely written as a sum

$$ F(x) = \alpha_{1} F_{1}(x) + \alpha_{2} F_{2}(x) + \alpha_{3} F_{3}(x), $$

where $\alpha_{1}, \alpha_{2}, \alpha_{3}$ are non-negative numbers with sum equal to 1, and $F_{1}, F_{2}, F_{3}$ are distribution functions such that $F_{1}(x)$ is absolutely continuous,

$$ F_{1}(x) = \int\limits_{-\infty}^{x} p(z) dz, $$

$F_{2}(x)$ is a "step-function",

$$ F_{2}(x) = \sum\limits_{x_{k} < x} p_{k}, $$

where the $x_{k}$ are the points where $F(x)$ "jumps" and the $p_{k} > 0$ are proportional to the size of these jumps, and $F_{3}(x)$ is the "singular" component — a continuous function whose derivative is zero almost-everywhere.


Example. Let $X_{k}$, $k = 1, 2, \ldots,$ be an infinite sequence of independent random variables assuming the values 1 and 0 with probabilities $0 < p_{k} \le \frac{1}{2}$ and $q_{k} = 1 - p_{k}$, respectively. Also, let

$$ X = \sum\limits_{k = 1}^{\infty} \frac{X_{k}}{2^{k}} $$

Now:

1) if $p_k = q_k = \frac{1}{2}$ for all $k$, then $X$ has an absolutely-continuous distribution function (with $p(x) = 1$ for $0 \le x \le 1$, that is, $X$ is uniformly distributed on $\left[ 0, 1 \right]$);

2) if $\sum\limits_{k = 1}^{\infty} p_k < \infty$, then $X$ has a "step" distribution function (it has jumps at all the dyadic-rational points in $\left[ 0, 1 \right]$);

3) if $\sum\limits_{k = 1}^{\infty} p_k = \infty$ and $p_k \rightarrow 0$ as $k \rightarrow \infty$, then $X$ 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 $P$ and $Q$ on the real line is often defined in terms of the corresponding distribution functions $F$ and $S$, by putting, for example,

$$ \rho_1(P, Q) = \sup_{x} \left| F(x) - S(x) \right| $$

or

$$ \rho_2(P, Q) = \mathrm{Var} \left( F(x) - S(x) \right) $$

(see Distributions, convergence of; Lévy metric; Characteristic function).

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 $F$ using results of independent observations, one can use some measure of the deviation of $F$ 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.

References

[C] H. Cramér, "Random variables and probability distributions", Cambridge Univ. Press (1970) MR0254895 Zbl 0184.40101
[C2] H. Cramér, "Mathematical methods of statistics", Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014
[F] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1957–1971)
[BS] 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) Zbl 0529.62099

Comments

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 $X$ is the function $F(x) = \mathrm{P} \lbrace X \le x \rbrace$. It then has the properties 1); 2') $F(x)$ is right-continuous at every $x$; 3). The unique probability distribution $P_{F}$ corresponding to it is now defined as

$$ P_{F}(a, b) = F(b) - F(a) $$

while the "step-function" $F_{2}(x)$ in the above-mentioned decomposition $F = \alpha_{1} F_{1} + \alpha_{2} F_{2} + \alpha_{3} F_{3} $ is

$$ F_{2} (x) = \sum\limits_{x_{k} \le x} p_{k}. $$

References

[JK] N.L. Johnson, S. Kotz, "Distributions in statistics" , Houghton Mifflin (1970) MR0270476 MR0270475 Zbl 0213.21101
How to Cite This Entry:
Distribution function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_function&oldid=30008
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article