Uniform distribution

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

For example, the results of the six possible ocular figures at dice after a roll: {1, 2, 3, 4, 5, 6}. In an ideal cube is the probability of occurrence of each of these values, 1/6, since it is the same for each possible value and must result in the sum of the individual probabilities 1.

Definition

Discrete event

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

Continuous case

Be a finite real interval, ie for. The probability of an event is defined in a uniform distribution as

Where the Lebesgue measure called. In particular, for a sub-interval

The probability density function here is a piecewise constant function with:

Using the indicator function of the interval is shorter writes this in the form

Similarly, you can declare a continuous uniform distribution on bounded subsets of the -dimensional space. For an event to get to the one-dimensional case analogous formula

Where the -dimensional Lebesgue measure called.

Examples

• When cubes of an ideal cube is the probability of each eye number between one and six equal to 1 / 6th
• The toss of a coin is the ideal probability for each of the two sides equal to 1/2.

Laplace

The uniform distribution was research area for Pierre -Simon Laplace, who suggested that if you do not know the probability measure on a probability space, first of all should accept equal distribution ( principle of indifference ). According to him, it is called a probability space for finite Ω and Laplace space.