equal distribution

This article deals with the mathematical uniform distribution. For the economic equal distribution see the Gini coefficient .

The term uniform distribution comes from probability theory and describes a probability distribution with certain properties. In the discrete case , every possible result occurs with the same probability , in the continuous case the density is constant. The basic idea of ​​equal distribution is that there is no preference.

For example, the results of the dice after a toss the six possible eye numbers: . In the case of an ideal cube, the probability of occurrence of each of these values ​​is 1/6, since it is the same for each of the six possible values ​​and the sum of the individual probabilities must be 1. ${\displaystyle \Omega =\{1,2,3,4,5,6\}}$

definition

Discreet case

→ Main article : Discrete equal distribution

Let be a non-empty finite set. Then, in a uniform distribution is the probability of an event with by the Laplace formula defined: ${\displaystyle \Omega }$ ${\displaystyle P(A)}$ ${\displaystyle A}$ ${\displaystyle A\subseteq \Omega }$

${\displaystyle P(A)={\frac {|A|}{|\Omega |}}={\frac {{\text{Anzahl der Elemente von }}A}{{\text{Anzahl der Elemente von }}\Omega }}}$

Steady fall

→ Main article : Constant equal distribution

Let be a finite real interval , i.e. for . With an even distribution, the probability of an event is defined as ${\displaystyle \Omega }$ ${\displaystyle \Omega =[a,b]}$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle A\subseteq \Omega }$

${\displaystyle P(A)=\int _{A}{\frac {1}{\lambda (\Omega )}}\,\mathrm {d} \lambda (x)={\frac {\lambda (A)}{\lambda (\Omega )}}={\frac {\lambda (A)}{b-a}},}$

where denotes the Lebesgue measure . In particular applies to a sub-interval ${\displaystyle \lambda }$ ${\displaystyle A=[c,d]\subseteq [a,b]}$

${\displaystyle P(A)={\frac {\lambda (A)}{\lambda (\Omega )}}={\frac {d-c}{b-a}}.}$

The probability density function here is a piecewise constant function with: ${\displaystyle \rho }$

${\displaystyle \rho (x)={\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b,\\0&{\text{sonst}}.\end{cases}}}$

With the help of the indicator function of the interval , this is written in a shorter form ${\displaystyle [a,b]}$

${\displaystyle \rho (x)={\frac {1}{b-a}}\cdot \mathbf {1} _{[a,b]}(x).}$

In a similar way, a constant uniform distribution can also be explained on limited subsets of the -dimensional space . For an event , the formula analogous to the one-dimensional case is obtained ${\displaystyle \Omega }$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A\subseteq \Omega }$

${\displaystyle P(A)=\int _{A}{\frac {1}{\lambda ^{n}(\Omega )}}\,\mathrm {d} \lambda ^{n}(x)={\frac {\lambda ^{n}(A)}{\lambda ^{n}(\Omega )}},}$

where the -dimensional Lebesgue measure denotes. ${\displaystyle \lambda ^{n}}$ ${\displaystyle n}$

Examples

• When throwing an ideal dice , the probability of any number between 1 and 6 being 1/6.
• When tossing an ideal coin, the probability for either side is 1/2.

Laplace's principle of indifference and equal distribution

Uniform distribution was a research area for Pierre-Simon Laplace , who suggested that one should first assume uniform distribution if one does not know the probability measure in a probability space ( principle of indifference ). According to him, a probability space for finite Ω is also called Laplace space. ${\displaystyle (\Omega ,\,{\mathfrak {P}}(\Omega ),\,{\mathcal {U}}_{\Omega })}$

See also

• Law of Small Numbers
• Uniform distribution modulo 1

Web links

• AV Prokhorov: Uniform distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

• Eric W. Weisstein : Uniform Distribution . In: MathWorld (English).

literature

• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , doi : 10.1007 / 978-3-663-09885-0 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 22.