HOME

TheInfoList



OR:

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and statistics, the discrete uniform distribution is a
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
wherein 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
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ...
. The possible values are 1, 2, 3, 4, 5, 6, and each time the die 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 because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
. For instance, a
random permutation A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and sim ...
is a
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
generated uniformly from the permutations of a given length, and a uniform spanning tree is a
spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...
generated uniformly from the spanning trees of a given graph. 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 ,''n''with the single parameter ''n''). With these conventions, the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
(CDF) of the discrete uniform distribution can be expressed, for any ''k'' ∈ 'a'',''b'' as :F(k;a,b)=\frac


Estimation of maximum

This example is described by saying that a sample of ''k'' observations is obtained from a uniform distribution on the integers 1,2,\dotsc,N, 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 World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
. The uniformly minimum variance unbiased (UMVU) estimator for the maximum is given by :\hat=\frac m - 1 = m + \frac - 1 where ''m'' is the sample maximum and ''k'' is the sample size, sampling without replacement. This can be seen as a very simple case of maximum spacing estimation. This has a variance of :\frac\frac \approx \frac \text k \ll N so a standard deviation of approximately \tfrac N k, the (population) average size of a gap between samples; compare \tfrac above. The sample maximum is the
maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed sta ...
estimator for the population maximum, but, as discussed above, it is biased. If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method.


Random permutation

See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed
random permutation A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and sim ...
.


Properties

The family of uniform distributions over ranges of integers (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, because 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.


See also

* Dirac delta distribution * Continuous uniform distribution


References

{{Probability distributions, discrete-finite Discrete distributions Location-scale family probability distributions su:Sebaran seragam#Kasus diskrit