Editing Maximum likelihood estimation (section)

=== Discrete uniform distribution ===
{{Main|German tank problem}}
Consider a case where ''n'' tickets numbered from 1 to ''n'' are placed in a box and one is selected at random (''see [[Uniform distribution (discrete)|uniform distribution]]''); thus, the sample size is 1.  If ''n'' is unknown, then the maximum likelihood estimator <math>\widehat{n}</math> of ''n'' is the number ''m'' on the drawn ticket. (The likelihood is 0 for ''n''&nbsp;<&nbsp;''m'', {{frac|1|''n''}} for ''n''&nbsp;≥&nbsp;''m'', and this is greatest when ''n''&nbsp;=&nbsp;''m''. Note that the maximum likelihood estimate of ''n'' occurs at the lower extreme of possible values {''m'',&nbsp;''m''&nbsp;+&nbsp;1,&nbsp;...}, rather than somewhere in the "middle" of the range of possible values, which would result in less bias.) The [[expected value]] of the number ''m'' on the drawn ticket, and therefore the expected value of <math>\widehat{n}</math>, is (''n''&nbsp;+&nbsp;1)/2. As a result, with a sample size of 1, the maximum likelihood estimator for ''n'' will systematically underestimate ''n'' by (''n''&nbsp;−&nbsp;1)/2.