Uniform distribution (discrete)
In probability theory and statistics, the discrete uni-form
distribution is a symmetric probability distribu-tion whereby a
nite number of values are equally likelyto be observed; every one
of n values has equal probabil-ity 1/n. Another way of saying
discrete uniform distri-bution would be a known, nite number of
outcomesequally likely to happen.A simple example of the discrete
uniform distribution isthrowing a fair die. The possible values are
1, 2, 3, 4,5, 6, and each time the die is thrown the probability of
agiven score is 1/6. If two dice are thrown and their val-ues
added, the resulting distribution is no longer uniformsince not all
sums have equal probability.The discrete uniform distribution
itself is inherently non-parametric. It is convenient, however, to
represent its val-ues generally by an integer interval [a,b], so
that a,b be-come the main parameters of the distribution (often
onesimply considers the interval [1,n] with the single param-eter
n). With these conventions, the cumulative distribu-tion function
(CDF) of the discrete uniform distributioncan be expressed, for any
k [a,b], as
F (k; a; b) =bkc a+ 1b a+ 1
1 Estimation of maximumMain article: German tank problem
This example is described by saying that a sample of k
ob-servations is obtained from a uniform distribution on
theintegers 1; 2; : : : ; N , with the problem being to estimatethe
unknown maximum N. This problem is commonlyknown as the German tank
problem, following the appli-cation of maximum estimation to
estimates of Germantank production during World War II.The UMVU
estimator for the maximum is given by
N^ =k + 1
km 1 = m+ m
k 1
wherem is the sample maximum and k is the sample size,sampling
without replacement.[1][2] This can be seen as avery simple case of
maximum spacing estimation.The formula may be understood
intuitively as:
The sample maximum plus the average gapbetween observations in
the sample,
the gap being added to compensate for the negative bias ofthe
sample maximum as an estimator for the populationmaximum.[notes
1]
This has a variance of[1]
1
k
(N k)(N + 1)(k + 2)
N2
k2samples small for k N
so a standard deviation of approximately Nk , the (popu-lation)
average size of a gap between samples; comparemk above.The sample
maximum is the maximum likelihood es-timator for the population
maximum, but, as discussedabove, it is biased.If samples are not
numbered but are recognizable ormarkable, one can instead estimate
population size viathe capture-recapture method.
2 Random permutationMain article: Random permutation
See rencontres numbers for an account of the
probabilitydistribution of the number of xed points of a
uniformlydistributed random permutation.
3 See also
Delta distribution
Uniform distribution (continuous)
4 Notes
[1] The sample maximum is never more than the populationmaximum,
but can be less, hence it is a biased estimator:it will tend to
underestimate the population maximum.
1
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