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discrete uniform distribution wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 estimation of maximum 2 random permutation 3 properties 4 see also 5 references toggle the table of contents discrete uniform distribution 32 languages العربية asturianu беларуская català deutsch ελληνικά esperanto español euskara فارسی suomi français galego עברית magyar italiano 日本語 한국어 македонски nederlands norsk bokmål polski русский slovenščina shqip српски srpski sunda ไทย türkçe українська 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia probability distribution on equally likely outcomes this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources discrete uniform distribution news newspapers books scholar jstor october 2022 learn how and when to remove this message discrete uniform probability mass function n 5 where n b a 1 cumulative distribution function notation u a b displaystyle mathcal u a b or u n i f a b displaystyle mathrm unif a b parameters a b displaystyle a b integers with b a displaystyle b geq a n b a 1 displaystyle n b a 1 support k a a 1 b 1 b displaystyle k in a a 1 dots b 1 b pmf 1 n displaystyle frac 1 n cdf k a 1 n displaystyle frac lfloor k rfloor a 1 n mean a b 2 displaystyle frac a b 2 median a b 2 displaystyle frac a b 2 mode n a variance b a 1 2 1 12 displaystyle frac b a 1 2 1 12 skewness 0 displaystyle 0 excess kurtosis 6 n 2 1 5 n 2 1 displaystyle frac 6 n 2 1 5 n 2 1 entropy ln n displaystyle ln n mgf e a t e b 1 t n 1 e t displaystyle frac e at e b 1 t n 1 e t cf e i a t e i b 1 t n 1 e i t displaystyle frac e iat e i b 1 t n 1 e it pgf z a z b 1 n 1 z displaystyle frac z a z b 1 n 1 z in probability theory and statistics the discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number n of outcome values are equally likely to be observed thus every one of the n outcome values has equal probability 1 n intuitively a discrete uniform distribution is a known finite number of outcomes all equally likely to happen a simple example of the discrete uniform distribution comes from throwing a fair six sided die the possible values are 1 2 3 4 5 6 and each time the die is thrown the probability of each given value is 1 6 if two dice were thrown and their values added the possible sums would not have equal probability and so the distribution of sums of two dice rolls is not uniform although it is common to consider discrete uniform distributions over a contiguous range of integers such as in this six sided die example one can define discrete uniform distributions over any finite set for instance the six sided die could have abstract symbols rather than numbers on each of its faces less simply a random permutation is a permutation generated uniformly randomly from the permutations of a given set and a uniform spanning tree of a graph is a spanning tree selected with uniform probabilities from the full set of spanning trees of the graph the discrete uniform distribution itself is non parametric however in the common case that its possible outcome values are the integers in an interval a b textstyle a b then a and b are parameters of the distribution and n b a 1 textstyle n b a 1 in these cases the cumulative distribution function cdf of the discrete uniform distribution can be expressed for any k as f k a b min max k a 1 b a 1 0 1 displaystyle f k a b min left max left frac lfloor k rfloor a 1 b a 1 0 right 1 right or simply f k a b k a 1 b a 1 displaystyle f k a b frac lfloor k rfloor a 1 b a 1 on the distribution s support k a b textstyle k in a b estimation of maximum edit main article german tank problem the problem of estimating the maximum n displaystyle n of a discrete uniform distribution on the integer interval 1 n displaystyle 1 n from a sample of k observations is commonly known as the german tank problem following the practical application of this maximum estimation problem during world war ii by allied forces seeking to estimate german tank production a uniformly minimum variance unbiased umvu estimator for the distribution s maximum in terms of m the sample maximum and k the sample size is 1 n k 1 k m 1 m m k 1 displaystyle hat n frac k 1 k m 1 m frac m k 1 this can be seen as a very simple case of maximum spacing estimation this has a variance of 1 1 k n k n 1 k 2 n 2 k 2 for small samples k n displaystyle frac 1 k frac n k n 1 k 2 approx frac n 2 k 2 text for small samples k ll n so a standard deviation of approximately n k displaystyle tfrac n k the population average gap size between samples the sample maximum m displaystyle m itself is the maximum likelihood estimator for the population maximum but it is biased if samples from a discrete uniform distribution are not numbered in order but are recognizable or markable one can instead estimate population size via a mark and recapture method random permutation edit 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 edit the family of uniform discrete 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 uniform discrete distributions over bounded integer ranges do not constitute an exponential family of distributions because their support varies with their parameters for families of distributions in which their supports do not depend on their parameters the pitman koopman darmois theorem states that only exponential families have sufficient statistics of dimensions that are bounded as sample size increases the uniform distribution is thus a simple example showing the necessity of the conditions for this theorem see also edit dirac delta distribution continuous uniform distribution references edit 1 2 johnson roger 1994 estimating the size of a population teaching statistics 16 2 summer 50 52 citeseerx 10 1 1 385 5463 doi 10 1111 j 1467 9639 1994 tb00688 x citation cite uses deprecated parameter citeseerx help v t e probability distributions list discrete univariate with finite support benford bernoulli beta binomial binomial categorical hypergeometric negative poisson binomial rademacher soliton discrete uniform zipf zipf mandelbrot with infinite support beta negative binomial borel conway maxwell poisson discrete phase type delaporte extended negative binomial flory schulz gauss kuzmin geometric logarithmic mixed poisson negative binomial panjer parabolic fractal poisson skellam yule simon zeta continuous univariate supported on a bounded interval arcsine argus balding nichols bates beta generalized beta rectangular continuous bernoulli continuous binomial irwin hall kumaraswamy logit normal noncentral beta pert power function raised cosine reciprocal triangular u quadratic uniform wigner semicircle supported on a semi infinite interval benini benktander 1st kind benktander 2nd kind beta prime burr chi chi squared noncentral inverse scaled dagum davis erlang hyper exponential hyperexponential hypoexponential logarithmic f noncentral folded normal fréchet gamma generalized inverse gamma gompertz gompertz shifted half logistic half normal hotelling s t squared hartman watson inverse gaussian generalized kolmogorov lévy log cauchy log laplace log logistic log normal log t lomax matrix exponential maxwell boltzmann maxwell jüttner mittag leffler nakagami pareto phase type poly weibull rayleigh relativistic breit wigner rice truncated normal type 2 gumbel weibull discrete wilks s lambda supported on the whole real line cauchy exponential power fisher s z kaniadakis κ gaussian gaussian q generalized hyperbolic generalized logistic logistic beta generalized normal geometric stable gumbel holtsmark hyperbolic secant johnson s s u landau laplace asymmetric logistic noncentral t normal gaussian normal inverse gaussian skew normal slash stable student s t tracy widom variance gamma voigt with support whose type varies generalized chi squared generalized extreme value generalized pareto marchenko pastur kaniadakis κ exponential kaniadakis κ gamma kaniadakis κ weibull kaniadakis κ logistic kaniadakis κ erlang q exponential q gaussian q weibull shifted log logistic tukey lambda mixed univariate continuous discrete rectified gaussian multivariate joint discrete ewens multinomial dirichlet negative continuous dirichlet generalized multivariate laplace multivariate normal multivariate stable multivariate t normal gamma inverse matrix valued lkj matrix beta matrix f matrix normal matrix t matrix gamma inverse wishart normal inverse normal inverse complex uniform distribution on a stiefel manifold directional univariate circular directional circular uniform univariate von mises wrapped normal wrapped cauchy wrapped exponential wrapped asymmetric laplace wrapped lévy bivariate spherical kent bivariate toroidal bivariate von mises multivariate von mises fisher bingham degenerate and singular degenerate dirac delta function singular cantor families circular compound poisson elliptical exponential natural exponential location scale maximum entropy mixture pearson tweedie wrapped category commons retrieved from https en wikipedia org w index php title discrete_uniform_distribution oldid 1320445563 categories discrete distributions location scale family probability distributions hidden categories articles with short description short description matches wikidata articles needing additional references from october 2022 all articles needing additional references cs1 errors deprecated parameters this page was last edited on 4 november 2025 at 18 52 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the 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