Uniform distribution: Information from Answers.com

discrete uniform
Probability mass function n = 5 where n = b − a + 1
Cumulative distribution function
Parameters	$a \in (\dots,-2,-1,0,1,2,\dots)\,$ $b \in (\dots,-2,-1,0,1,2,\dots)\,$ $n=b-a+1\,$
Support	$k \in \{a,a+1,\dots,b-1,b\}\,$
Probability mass function (pmf)	$\begin{matrix} \frac{1}{n} & \mbox{for }a\le k \le b\ \\0 & \mbox{otherwise } \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }k<a\\ \frac{\lfloor k \rfloor -a+1}{n} & \mbox{for }a \le k \le b \\1 & \mbox{for }k>b \end{matrix}$
Mean	$\frac{a+b}{2}\,$
Median	$\frac{a+b}{2}\,$
Mode	N/A
Variance	$\frac{(b-a+1)^2-1}{12}=\frac{n^2-1}{12},$
Skewness	$0\,$
Excess kurtosis	$-\frac{6(n^2+1)}{5(n^2-1)}\,$
Entropy	$\ln(n)\,$
Moment-generating function (mgf)	$\frac{e^{at}-e^{(b+1)t}}{n(1-e^t)}\,$
Characteristic function	$\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}$

In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable.

If a random variable has any of $n$ possible values $k_1,k_2,\dots,k_n$ that are equally probable, then it has a discrete uniform distribution. The probability of any outcome $k i$ is $1 / n$ . A simple example of the discrete uniform distribution is throwing a fair die. The possible values of $k$ 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, then the uniform distribution no longer fits, as values from 2 to 12 have varying probabilities.

In case the values of a random variable with a discrete uniform distribution are real, it is possible to express the cumulative distribution function in terms of the degenerate distribution; thus

$F(k;a,b,n)={1\over n}\sum_{i=1}^n H(k-k_i)$

where the Heaviside step function $H (x - x 0)$ is the CDF of the degenerate distribution centered at $x 0$ . This assumes that consistent conventions are used at the transition points.

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

Estimation of maximum

Main article: German tank problem

This example is described by saying that a sample of k observations are 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

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

^ 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

^ ^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, http://www.rsscse.org.uk/ts/gtb/johnson.pdf

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · 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]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

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

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

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

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

v • d • e Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • 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

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Uniform distribution

Contents

Estimation of maximum

Random permutation

See also

Notes

References