In probability theory and statistics, the discrete uniform distribution is a symmetricprobability distribution whereby a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".
A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the dice is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform since not all sums have equal probability.
The discrete uniform distribution itself is inherently non-parametric. It is convenient, however, to represent its values generally by all integers in an interval [a,b], so that a and b become the main parameters of the distribution (often one simply considers the interval [1,n] with the single parameter n). With these conventions, the cumulative distribution function (CDF) of the discrete uniform distribution can be expressed, for any k ∈ [a,b], as
This example is described by saying that a sample of k observations is obtained from a uniform distribution on the integers , with the problem being to estimate the unknown maximum N. This problem is commonly known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II.
For any integer m such that k ≤ m ≤ N, the probability that the sample maximum will be equal to m can be computed as follows. The number of different groups of k tanks that can be made from a total of N tanks is given by the binomial coefficient. Since in this way of counting the permutations of tanks are counted only once, we can order the serial numbers and take note of the maximum of each sample. To compute the probability we have to count the number of ordered samples that can be formed with the last element equal to m and all the other k-1 tanks less or equal to m-1. The number of samples of k-1 tanks that can be made from a total m-1 tanks is given by the binomial coefficient, so the probability of having a maximum m is .
Given the total number N and the sample size k, the expected value of the sample maximum is
The family of uniform distributions (with one or both bounds unknown) has a finite-dimensional sufficient statistic, namely the triple of the sample maximum, sample minimum, and sample size, but is not an exponential family of distributions, since the support varies with the parameters. For families whose support does not depend on the parameters, the Pitman–Koopman–Darmois theorem states that only exponential families have a sufficient statistic whose dimension is bounded as sample size increases. The uniform distribution is thus a simple example showing the limit of this theorem.