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.
Let be a finite real interval , i.e. for . With an even distribution, the probability of an event is defined as
where denotes the Lebesgue measure . In particular applies to a sub-interval
The probability density function here is a piecewise constant function with:
With the help of the indicator function of the interval , this is written in a shorter form
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
where the -dimensional Lebesgue measure denotes.
- 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.
- AV Prokhorov: Uniform distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- 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 .
- Georgii: Stochastics. 2009, p. 22.