Learn and talk about Uniform distribution (discrete), Discrete distributions

discrete uniform
Probability mass function n = 5 where n = b − a + 1
Cumulative distribution function
Parameters	$a \in (\dots,-2,-1,0,1,2,\dots)\,$ $b \in (\dots,-2,-1,0,1,2,\dots), b \ge a$ $n=b-a+1\,$
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	$\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}$

In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution whereby 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 since not all sums have equal probability.

The discrete uniform distribution itself is inherently non-parametric. It is convenient, however, to represent its values generally by an integer interval [a,b], so that a,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

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

1 Estimation of maximum
2 Random permutation
3 See also
4 Notes
5 References

Estimation of maximum[edit]

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 $1,2,\dots,N$ , 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 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]^[2] This can be seen as a very simple case of maximum spacing estimation.

The formula may be understood intuitively as:

"The sample maximum plus the average gap between observations in the sample",

the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.^{[notes 1]}

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

Notes[edit]

^ The sample maximum is never more than the population maximum, but can be less, hence it is a biased estimator: it will tend to underestimate the population maximum.

References[edit]

^ ^a ^b Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics 16 (2 (Summer)), doi:10.1111/j.1467-9639.1994.tb00688.x
^ Johnson, Roger (2006), "Estimating the Size of a Population", Getting the Best from Teaching Statistics

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

v t e Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Uniform_distribution_(discrete) — Please support Wikipedia.
A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

Uniform distribution (discrete)

Contents

Estimation of maximum[edit]

Random permutation[edit]

See also[edit]

Notes[edit]

References[edit]

Talk About Uniform distribution (discrete)

Support Wikipedia