Discrete uniform distribution

<h2 id="estimation-of-maximum">Estimation of maximum</h2>
<p class="note">Main article: <a href="/facts/German_tank_problem/xMSy97YB">German tank problem</a></p>
<p>The problem of estimating the maximum 
  
    
      
        N
      
    
    {\displaystyle N}
  
 of a discrete uniform distribution on the integer interval 
  
    
      
        [
        1
        ,
        N
        ]
      
    
    {\displaystyle [1,N]}
  
 from a sample of <i>k</i> observations is commonly known as the <a href="/facts/German_tank_problem/xMSy97YB">German tank problem</a>, following the practical application of this maximum estimation problem, during <a href="/facts/World_War_II/2efV5L1U">World War II</a>, by Allied forces seeking to estimate German tank production.
</p><p>A <a href="/facts/Uniformly_minimum_variance_unbiased_estimator/t4JhrJ3a">uniformly minimum variance unbiased</a> (UMVU) estimator for the distribution's maximum in terms of <i>m,</i> the <a href="/facts/Sample_maximum/8XmNUxyt">sample maximum</a>, and <i>k,</i> the <a href="/facts/Sample_size/sf8dAFzq">sample size</a>, is<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>
  
    
      
        
          
            
              N
              ^
            
          
        
        =
        
          
            
              k
              +
              1
            
            k
          
        
        m
        −
        1
        =
        m
        +
        
          
            m
            k
          
        
        −
        1.
      
    
    {\displaystyle {\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1.}

</p><p>This can be seen as a very simple case of <a href="/facts/Maximum_spacing_estimation/fuJj0qan">maximum spacing estimation</a>.
</p><p>This has a variance of<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>

1
            k
          
        
        
          
            
              (
              N
              −
              k
              )
              (
              N
              +
              1
              )
            
            
              (
              k
              +
              2
              )
            
          
        
        ≈
        
          
            
              N
              
                2
              
            
            
              k
              
                2
              
            
          
        
        
           for small samples 
        
        k
        ≪
        N
      
    
    {\displaystyle {\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
            
          
        
      
    
    {\displaystyle {\tfrac {N}{k}}}
  
, the population-average gap size between samples.
</p><p>The sample maximum 
  
    
      
        m
      
    
    {\displaystyle m}
  
 itself is the <a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">maximum likelihood estimator</a> for the population maximum, but it is biased.
</p><p>If samples from a discrete uniform distribution are not numbered in order but are recognizable or markable, one can instead estimate population size via a <a href="/facts/Mark_and_recapture/BcUXVJ3u">mark and recapture</a> method.
</p>
<h2 id="random-permutation">Random permutation</h2>
<p class="note">Main article: <a href="/facts/Random_permutation/zJMPNXlz">Random permutation</a></p>
<p>See <a href="/facts/Rencontres_numbers/S04j0k5r">rencontres numbers</a> for an account of the probability distribution of the number of fixed points of a uniformly distributed <a href="/facts/Random_permutation/zJMPNXlz">random permutation</a>.
</p>
<h2 id="properties">Properties</h2>
<p>The family of uniform discrete distributions over ranges of integers with one or both bounds unknown has a finite-dimensional <a href="/facts/Sufficient_statistic/eClRXHpd">sufficient statistic</a>, namely the triple of the sample maximum, sample minimum, and sample size.
</p><p>Uniform discrete distributions over bounded integer ranges do not constitute an <a href="/facts/Exponential_family/1LkkqEIf">exponential family</a> of distributions because their <a href="/facts/Support_of_a_distribution/BTuMGbsb">support</a> varies with their parameters.
</p><p>For families of distributions in which their supports do not depend on their parameters, the <a href="/facts/Pitman%25E2%2580%2593Koopman%25E2%2580%2593Darmois_theorem/1LkkqEIf">Pitman–Koopman–Darmois theorem</a> states that only exponential families have sufficient statistics of dimensions that are bounded as sample size increases. The uniform distribution is thus a simple example showing the necessity of the conditions for this theorem.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dirac_delta_distribution/V8PjRP8T">Dirac delta distribution</a></li>
<li><a href="/facts/Continuous_uniform_distribution/XbnlVljT">Continuous uniform distribution</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics, 16 (2 (Summer)): 50–52, CiteSeerX 10.1.1.385.5463, doi:10.1111/j.1467-9639.1994.tb00688.x <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics, 16 (2 (Summer)): 50–52, CiteSeerX 10.1.1.385.5463, doi:10.1111/j.1467-9639.1994.tb00688.x <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Discrete uniform distribution open-in-new

Discrete uniform distribution