Fair dice are the textbook discrete uniform
A discrete uniform distribution gives each of n known finite outcomes the same probability 1/n. A fair six-sided die—faces 1 through 6, each with chance 1/6—is the classroom example; sums of two dice are not uniform because totals have unequal odds.
The support need not be consecutive integers: abstract die faces, a uniformly random permutation, or a uniformly chosen spanning tree of a graph all qualify. When outcomes are the integers from a through b, those bounds become parameters and the cumulative distribution function can be written with floor functions that rise in equal steps between the endpoints. The family over integer ranges with unknown bounds is non-parametric in a deep sense and is not an exponential family because the support itself moves with the parameters.
Estimating an unknown maximum from a sample—famous as the German tank problem when Allies inferred Nazi tank production from captured serial numbers—has a uniformly minimum-variance unbiased solution in terms of the sample maximum m and sample size k: roughly ((k+1)/k)m − 1. The sample maximum alone is the maximum-likelihood estimator yet biased low. Mark-and-recapture offers another population-size route when items can be tagged rather than serially numbered. Rencontres numbers describe how many fixed points a uniform random permutation tends to leave—another discrete-uniform offspring.
A finite-dimensional sufficient statistic exists for uniform integer ranges with unknown ends: the triple of sample minimum, sample maximum, and sample size. That bounded sufficiency while support depends on parameters is exactly why the Pitman–Koopman–Darmois theorem needs its support-fixed assumption—uniform discrete laws are the clean counter-example students meet right after learning fair dice. Intuition stays simple—known finite outcomes, each equally likely—yet inference about unknown bounds shows how much structure hides inside a fair die's cousin.
Source: Discrete uniform distribution
