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((Short description|Probability distribution on equally likely outcomes))
((More citations needed|date=October 2022))
<!--- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one --->
((Probability distribution|
  name       =discrete uniform|
  type       =mass|
  pdf_image  =[[Image:Uniform discrete pmf svg.svg|325px|Discrete uniform probability mass function for ''n'' = 5]]<br /><small>''n'' = 5 where ''n'' = ''b''&nbsp;&minus;&nbsp;''a''&nbsp;+&nbsp;1</small>|
  cdf_image  =[[Image:Dis Uniform distribution CDF.svg|325px|Discrete uniform cumulative distribution function for ''n'' = 5]]<br /><small></small>|
  notation   = <math>\mathcal{U}\{a, b\}</math>  or  <math>\mathrm{unif}\{a,b\}</math>|
  parameters =<math>a,b</math> integers with <math>b \geq a</math><br /><math>n=b-a+1</math>|
  support    =<math>k \in \{a,a+1,\dots,b-1,b\}</math>|
  pdf        =<math>\frac{1}{n}</math>|
  cdf        =<math> \frac{\lfloor k \rfloor -a+1}{n} </math>|
  mean       =<math>\frac{a+b}{2}</math>|
  median     =<math>\frac{a+b}{2}</math>|
  mode       =N/A|
  variance   =<math>\frac{(b-a+1)^2-1}{12}</math>|
  skewness   =<math>0</math>|
  kurtosis   =<math>-\frac{6(n^2+1)}{5(n^2-1)}</math>|
  entropy    =<math>\ln(n)</math>|
  mgf        =<math>\frac{e^{at}-e^{(b+1)t)){n(1-e^t)}</math>|
  char       =<math>\frac{e^{iat}-e^{i(b+1)t)){n(1-e^{it})}</math>|
  pgf        =<math>\frac{z^{a}-z^{b+1)){n(1-z)}</math>|
))

In [[probability theory]] and [[statistics]], the '''discrete uniform distribution''' is a [[Symmetric distribution|symmetric]] [[discrete probability distribution|probability distribution]] 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|die]]. 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]]. For instance, a [[random permutation]] is a [[permutation]] generated uniformly from the permutations of a given length, and a [[uniform spanning tree]] is a [[spanning tree]] 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 [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

:<math>F(k;a,b)=\frac{\lfloor k \rfloor -a + 1}{b-a+1}</math>

==Estimation of maximum==
((main article|German tank problem))
This example is described by saying that a sample of ''k'' observations is obtained from a uniform distribution on the integers <math>1,2,\dotsc,N</math>, 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]].

The [[Uniformly minimum variance unbiased estimator|uniformly minimum variance unbiased]] (UMVU) estimator for the maximum is given by

:<math>\hat{N}=\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1</math>
where ''m'' is the [[sample maximum]] and ''k'' is the [[sample size]], sampling without replacement.<ref name="Johnson">((citation|last=Johnson|first=Roger|title=Estimating the Size of a Population|journal=Teaching Statistics|volume=16|issue=2 (Summer)|pages=50–52|year=1994|doi=10.1111/j.1467-9639.1994.tb00688.x|citeseerx=10.1.1.385.5463))</ref> This can be seen as a very simple case of [[maximum spacing estimation]].

This has a variance of<ref name="Johnson"/>
:<math>\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N</math>
so a standard deviation of approximately <math>\tfrac N k</math>, the (population) average size of a gap between samples; compare <math>\tfrac{m}{k}</math> above.

The sample maximum is the [[maximum likelihood]] 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==
((main article|Random permutation))
See [[rencontres numbers]] for an account of the probability distribution of the number of fixed points of a uniformly distributed [[random permutation]].

==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 of a distribution|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==
((reflist))

((Probability distributions|discrete-finite))

[[Category:Discrete distributions]]
[[Category:Location-scale family probability distributions]]

[[su:Sebaran seragam#Kasus diskrit]]

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