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| <!--- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one --->
| | == looked distant == |
| {{Probability distribution|
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| name =discrete uniform|
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| type =mass|
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| pdf_image =[[Image:Uniform discrete pmf svg.svg|325px|Discrete uniform probability mass function for ''n'' = 5]]<br /><small>''n'' = 5 where ''n'' = ''b'' − ''a'' + 1</small>|
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| cdf_image =[[Image:Dis Uniform distribution CDF.svg|325px|Discrete uniform cumulative distribution function for ''n'' = 5]]<br /><small></small>|
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| parameters =<math>a \in (\dots,-2,-1,0,1,2,\dots)\,</math><br /><math>b \in (\dots,-2,-1,0,1,2,\dots), b \ge a</math><br /><math>n=b-a+1\,</math>|
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| support =<math>k \in \{a,a+1,\dots,b-1,b\}\,</math>|
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| pdf =<math>\frac{1}{n}</math>|
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| cdf =<math> \frac{\lfloor k \rfloor -a+1}{n} </math>|
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| mean =<math>\frac{a+b}{2}\,</math>|
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| median =<math>\frac{a+b}{2}\,</math>|
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| mode =N/A|
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| variance =<math>\frac{n^2-1}{12}</math>|
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| skewness =<math>0\,</math>|
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| kurtosis =<math>-\frac{6(n^2+1)}{5(n^2-1)}\,</math>|
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| entropy =<math>\ln(n)\,</math>|
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| mgf =<math>\frac{e^{at}-e^{(b+1)t}}{n(1-e^t)}\,</math>|
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| char =<math>\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}</math>|
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| }}
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| In [[probability theory]] and [[statistics]], the '''discrete uniform distribution''' is a [[Symmetric distribution|symmetric]] [[discrete probability distribution|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".
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| | | 相关的主题文章： |
| A simple example of the discrete uniform distribution is throwing a fair {{dice}}. The possible values are 1, 2, 3, 4, 5, 6, and each time the {{dice}} 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.
| | <ul> |
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| 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
| | <li>[http://www.avbodo.com/thread-10186-1-1.html http://www.avbodo.com/thread-10186-1-1.html]</li> |
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| :<math>F(k;a,b)=\frac{\lfloor k \rfloor -a + 1}{b-a+1}</math>
| | <li>[http://www.baimusic.cn/forum.php?mod=viewthread&tid=160050 http://www.baimusic.cn/forum.php?mod=viewthread&tid=160050]</li> |
| | | |
| ==Estimation of maximum==
| | <li>[http://bt.aia.edu.cn/plus/feedback.php?aid=1 http://bt.aia.edu.cn/plus/feedback.php?aid=1]</li> |
| {{main|German tank problem}}
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| This example is described by saying that a sample of ''k'' observations is obtained from a uniform distribution on the integers <math>1,2,\dots,N</math>, 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]].
| | </ul> |
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| The [[UMVU]] estimator for the maximum is given by
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| :<math>\hat{N}=\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1</math>
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| where ''m'' is the [[sample maximum]] and ''k'' is the [[sample size]], sampling without replacement.<ref name="Johnson">{{citation
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| |last=Johnson
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| |first=Roger
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| |title=Estimating the Size of a Population
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| |year=1994
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| |journal=[http://www.rsscse.org.uk/ts/index.htm Teaching Statistics]
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| |volume=16
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| |issue=2 (Summer)
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| |doi=10.1111/j.1467-9639.1994.tb00688.x
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| }}</ref><ref name="Johnson2">{{citation
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| |last=Johnson
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| |first=Roger
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| |contribution=Estimating the Size of a Population
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| |title=Getting the Best from Teaching Statistics
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| |year=2006
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| |url=http://www.rsscse.org.uk/ts/gtb/contents.html
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| |contribution-url=http://www.rsscse.org.uk/ts/gtb/johnson.pdf
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| }}</ref> This can be seen as a very simple case of [[maximum spacing estimation]].
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| The formula may be understood intuitively as:
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| :"The sample maximum plus the average gap between observations in the sample",
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| the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.<ref group="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.</ref>
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| This has a variance of<ref name="Johnson"/>
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| :<math>\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N</math>
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| so a standard deviation of approximately <math>N/k</math>, the (population) average size of a gap between samples; compare <math>\frac{m}{k}</math> above.
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| The sample maximum is the [[maximum likelihood]] estimator for the population maximum, but, as discussed above, it is biased.
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| If samples are not numbered but are recognizable or markable, one can instead estimate population size via the [[capture-recapture]] method.
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| ==Random permutation== | |
| {{main|Random permutation}}
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| See [[rencontres numbers]] for an account of the probability distribution of the number of fixed points of a uniformly distributed [[random permutation]].
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| ==See also==
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| * [[Delta distribution]]
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| * [[Uniform distribution (continuous)]]
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| ==Notes==
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| {{reflist|group=notes}}
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| ==References==
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| {{reflist}}
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| {{ProbDistributions|discrete-finite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Uniform Distribution (Discrete)}}
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| [[Category:Discrete distributions]] | |
| [[Category:Probability distributions]]
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| [[su:Sebaran seragam#Kasus diskrit]]
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