Discrete uniform distribution

discrete uniform
	Probability mass function; n = 5 where n = b − a + 1
	Cumulative distribution function;
Notation	or
Parameters	integers with ;
Support
PMF
CDF
Mean
Median
Mode	N/A
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF
PGF

In probability theory and statistics, the discrete uniform distribution is a symmetric 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 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

F(k;a,b)={\frac {\lfloor k\rfloor -a+1}{b-a+1}}

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.

The uniformly minimum variance unbiased (UMVU) estimator for the maximum is given by

{\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1

where m is the sample maximum and k is the sample size, sampling without replacement.[1] This can be seen as a very simple case of maximum spacing estimation.

This has a variance of[1]

{\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 ${\tfrac {N}{k}}$ , the (population) average size of a gap between samples; compare ${\tfrac {m}{k}}$ 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

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 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.

References

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

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[Johnson-1] 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

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
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

discrete uniform
Probability mass function n = 5 where n = b − a + 1
Cumulative distribution function
Notation	${\mathcal {U}}\{a,b\}$ or $\mathrm {unif} \{a,b\}$
Parameters	$a,b$ integers with $b\geq a$ $n=b-a+1$
Support	$k\in \{a,a+1,\dots ,b-1,b\}$
PMF	${\frac {1}{n}}$
CDF	${\frac {\lfloor k\rfloor -a+1}{n}}$
Mean	${\frac {a+b}{2}}$
Median	${\frac {a+b}{2}}$
Mode	N/A
Variance	${\frac {n^{2}-1}{12}}$
Skewness	$0$
Ex. kurtosis	$-{\frac {6(n^{2}+1)}{5(n^{2}-1)}}$
Entropy	$\ln(n)$
MGF	${\frac {e^{at}-e^{(b+1)t}}{n(1-e^{t})}}$
CF	${\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}$
PGF	${\frac {z^{a}-z^{b+1}}{n(1-z)}}$

Discrete uniform distribution

Estimation of maximum

Random permutation

Properties

See also

References