Editing Binomial distribution (section)

=== Median ===
In general, there is no single formula to find the [[median]] for a binomial distribution, and it may even be non-unique. However, several special results have been established:
* If {{math|''np''}} is an integer, then the mean, median, and mode coincide and equal {{math|''np''}}.<ref>{{cite journal|last=Neumann|first=P.|year=1966|title=Über den Median der Binomial- and Poissonverteilung|journal=Wissenschaftliche Zeitschrift der Technischen Universität Dresden|volume=19|pages=29–33|language=de}}</ref><ref>Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", [[The Mathematical Gazette]] 94, 331-332.</ref>
* Any median {{math|''m''}} must lie within the interval <math>\lfloor np \rfloor\leq m \leq \lceil np \rceil</math>.<ref name="KaasBuhrman">{{cite journal|first1=R.|last1=Kaas|first2=J.M.|last2=Buhrman|title=Mean, Median and Mode in Binomial Distributions|journal=Statistica Neerlandica|year=1980|volume=34|issue=1|pages=13–18|doi=10.1111/j.1467-9574.1980.tb00681.x}}</ref>
* A median {{math|''m''}} cannot lie too far away from the mean:<math>|m-np|\leq \min\{{\ln2}, \max\{p,1-p\}\}</math> .<ref name="Hamza">
{{cite journal
 | last1 = Hamza | first1 = K.
 | doi = 10.1016/0167-7152(94)00090-U
 | title = The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions
 | journal = Statistics & Probability Letters
 | volume = 23
 | pages = 21–25
 | year = 1995
}}</ref>
* The median is unique and equal to {{math|1=''m'' = [[Rounding|round]](''np'')}} when {{math|1={{abs|''m'' − ''np''}} ≤ min{{brace|''p'', 1 − ''p''}}}} (except for the case when {{math|1=''p'' = 1/2}} and {{math|''n''}} is odd).<ref name="KaasBuhrman"/>
* When {{math|''p''}} is a rational number (with the exception of {{math|1=''p'' = 1/2}}\ and {{math|''n''}} odd) the median is unique.<ref name="Nowakowski">
{{cite journal
 | last1 = Nowakowski | first1 = Sz.
 | doi = 10.37418/amsj.10.4.9
 | issn=1857-8365
 | title = Uniqueness of a Median of a Binomial Distribution with Rational Probability
 | journal = Advances in Mathematics: Scientific Journal
 | volume = 10
 | issue = 4
 | pages = 1951–1958
 | year = 2021
 | arxiv = 2004.03280
 | s2cid = 215238991
}}</ref>
* When <math>p= \frac{1}{2} </math> and {{math|''n''}} is odd, any number {{math|''m''}} in the interval <math> \frac{1}{2} \bigl(n-1\bigr)\leq m \leq  \frac{1}{2}  \bigl(n+1\bigr)</math>  is a median of the binomial distribution. If <math>p= \frac{1}{2}  </math> and {{math|''n''}} is even, then  <math>m= \frac{n}{2} </math> is the unique median.

=== Tail bounds ===
For {{math|''k'' ≤ ''np''}}, upper bounds can be derived for the lower tail of the cumulative distribution function <math>F(k;n,p) = \Pr(X \le k)</math>, the probability that there are at most {{math|''k''}} successes. Since <math>\Pr(X \ge k) = F(n-k;n,1-p) </math>, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for {{math|''k'' ≥ ''np''}}.

[[Hoeffding's inequality]] yields the simple bound
: <math> F(k;n,p) \leq \exp\left(-2 n\left(p-\frac{k}{n}\right)^2\right), \!</math>
which is however not very tight. In particular, for {{math|1=''p'' = 1}}, we have that {{math|1=''F''(''k''; ''n'', ''p'') = 0}} (for fixed {{math|''k''}}, {{math|''n''}} with {{math|''k'' < ''n''}}), but Hoeffding's bound evaluates to a positive constant.

A sharper bound can be obtained from the [[Chernoff bound]]:<ref name="ag">{{cite journal |first1=R. |last1=Arratia |first2=L. |last2=Gordon |title=Tutorial on large deviations for the binomial distribution |journal=Bulletin of Mathematical Biology |volume=51 |issue=1 |year=1989 |pages=125–131 |doi=10.1007/BF02458840 |pmid=2706397 |s2cid=189884382 }}</ref>
: <math> F(k;n,p) \leq \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right)  </math>
where {{math|''D''(''a'' ∥ ''p'')}} is the [[Kullback–Leibler divergence|relative entropy (or Kullback-Leibler divergence)]] between an {{math|''a''}}-coin and a {{math|''p''}}-coin (i.e. between the {{math|Bernoulli(''a'')}} and {{math|Bernoulli(''p'')}} distribution):
: <math> D(a\parallel p)=(a)\ln\frac{a}{p}+(1-a)\ln\frac{1-a}{1-p}. \!</math>

Asymptotically, this bound is reasonably tight; see <ref name="ag"/> for details.

One can also obtain ''lower'' bounds on the tail {{math|''F''(''k''; ''n'', ''p'')}}, known as anti-concentration bounds. By approximating the binomial coefficient with [[Stirling's approximation|Stirling's formula]] it can be shown that<ref>{{cite book |author1=Robert B. Ash |title=Information Theory |url=https://archive.org/details/informationtheor00ashr |url-access=limited |date=1990 |publisher=Dover Publications |page=[https://archive.org/details/informationtheor00ashr/page/n81 115]|isbn=9780486665214 }}</ref>
: <math> F(k;n,p) \geq \frac{1}{\sqrt{8n\tfrac{k}{n}(1-\tfrac{k}{n})}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right),</math>
which implies the simpler but looser bound
: <math> F(k;n,p) \geq \frac1{\sqrt{2n}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right).</math>

For {{math|1=''p'' = 1/2}} and {{math|''k'' ≥ 3''n''/8}} for even {{math|''n''}}, it is possible to make the denominator constant:<ref>{{cite web |last1=Matoušek |first1=J. |last2=Vondrak |first2=J. |title=The Probabilistic Method |work=lecture notes |url=https://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f09/www/handouts/matousek-vondrak-prob-ln.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f09/www/handouts/matousek-vondrak-prob-ln.pdf |archive-date=2022-10-09 |url-status=live }}</ref>
: <math> F(k;n,\tfrac{1}{2}) \geq \frac{1}{15} \exp\left(- 16n \left(\frac{1}{2} -\frac{k}{n}\right)^2\right). \!</math>