# equal distribution

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

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}$

${\ displaystyle P (A) = {\ frac {| A |} {| \ Omega |}} = {\ frac {{\ text {number of elements of}} A} {{\ text {number of elements of} } \ Omega}}}$

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}$

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

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

${\ displaystyle P (A) = {\ frac {\ lambda (A)} {\ lambda (\ Omega)}} = {\ frac {dc} {ba}}.}$

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

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

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

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

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. ${\ displaystyle (\ Omega, \, {\ mathfrak {P}} (\ Omega), \, {\ mathcal {U}} _ {\ Omega})}$