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	~~Im Eduardo~~ and ~~was born on 12 March 1978~~. ~~My hobbies~~ are ~~Element collecting~~ and ~~Vehicle restoration~~.<br><br>~~Here~~ is ~~my homepage~~ ... ~~Fifa 15 Coin Generator~~ ([http://~~Www~~.~~sanyangyasuoji~~.~~com~~/plus/~~guestbook~~.~~php Www~~.~~Sanyangyasuoji~~.~~com~~])		<!--- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one --->
			{{Probability distribution\|
			name =discrete uniform\|
			type =mass\|
			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>\|
			cdf_image =[[Image:Dis Uniform distribution CDF.svg\|325px\|Discrete uniform cumulative distribution function for ''n'' = 5]]<br /><small></small>\|
			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>\|
			support =<math>k \in \{a,a+1,\dots,b-1,b\}\,</math>\|
			pdf =<math>\frac{1}{n}</math>\|
			cdf =<math> \frac{\lfloor k \rfloor -a+1}{n} </math>\|
			mean =<math>\frac{a+b}{2}\,</math>\|
			median =<math>\frac{a+b}{2}\,</math>\|
			mode =N/A\|
			variance =<math>\frac{n^2-1}{12}</math>\|
			skewness =<math>0\,</math>\|
			kurtosis =<math>-\frac{6(n^2+1)}{5(n^2-1)}\,</math>\|
			entropy =<math>\ln(n)\,</math>\|
			mgf =<math>\frac{e^{at}-e^{(b+1)t}}{n(1-e^t)}\,</math>\|
			char =<math>\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}</math>\|
			}}

			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".

			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.

			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

			:<math>F(k;a,b)=\frac{\lfloor k \rfloor -a + 1}{b-a+1}</math>

			==Estimation of maximum==
			{{main\|German tank problem}}
			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]].

			The [[UMVU]] estimator for the maximum is given by
			:<math>\hat{N}=\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1</math>
			where ''m'' is the [[sample maximum]] and ''k'' is the [[sample size]], sampling without replacement.<ref name="Johnson">{{citation
			\|last=Johnson
			\|first=Roger
			\|title=Estimating the Size of a Population
			\|year=1994
			\|journal=[http://www.rsscse.org.uk/ts/index.htm Teaching Statistics]
			\|volume=16
			\|issue=2 (Summer)
			\|doi=10.1111/j.1467-9639.1994.tb00688.x
			}}</ref><ref name="Johnson2">{{citation
			\|last=Johnson
			\|first=Roger
			\|contribution=Estimating the Size of a Population
			\|title=Getting the Best from Teaching Statistics
			\|year=2006
			\|url=http://www.rsscse.org.uk/ts/gtb/contents.html
			\|contribution-url=http://www.rsscse.org.uk/ts/gtb/johnson.pdf
			}}</ref> 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.<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>

			This has a variance of<ref name="Johnson"/>
			:<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>
			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.

			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\|Random permutation}}
			See [[rencontres numbers]] for an account of the probability distribution of the number of fixed points of a uniformly distributed [[random permutation]].

			==See also==
			* [[Delta distribution]]
			* [[Uniform distribution (continuous)]]

			==Notes==
			{{reflist\|group=notes}}

			==References==
			{{reflist}}

			{{ProbDistributions\|discrete-finite}}
			{{Common univariate probability distributions}}

			{{DEFAULTSORT:Uniform Distribution (Discrete)}}
			[[Category:Discrete distributions]]
			[[Category:Probability distributions]]

			[[su:Sebaran seragam#Kasus diskrit]]

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Im Eduardo and was born on 12 March 1978. My hobbies are Element collecting and Vehicle restoration.<br><br>Here is my homepage ... Fifa 15 Coin Generator ([http://Www.sanyangyasuoji.com/plus/guestbook.php Www.Sanyangyasuoji.com])

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