Uniform distribution (discrete)

In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable.

A random variable that has any of $$n$$ possible values $$k_1,k_2,\dots,k_n$$ that are equally probable, has a discrete uniform distribution, then the probability of any outcome $$k_i$$ is $$1/n$$. A simple example of the discrete uniform distribution is throwing a fair die. The possible values of $$k$$ are 1, 2, 3, 4, 5, 6; and each time the die is thrown, the probability of a given score is 1/6.

In case the values of a random variable with a discrete uniform distribution are real, it is possible to express the cumulative distribution function in terms of the degenerate distribution; thus


 * $$F(k;a,b,n)={1\over n}\sum_{i=1}^n H(k-k_i)$$

where the Heaviside step function $$H(x-x_0)$$ is the CDF of the degenerate distribution centered at $$x_0$$. This assumes that consistent conventions are used at the transition points.

See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation.

Diskrete Gleichverteilung Variabile casuale Uniforme discreta Discrete uniforme verdeling loi uniforme discrète