equal distribution

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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.


Discreet case

Let be a non-empty finite set. Then, in a uniform distribution is the probability of an event with by the Laplace formula defined:

Steady fall

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.

See also

Web links


Individual evidence

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