Editing Binomial distribution (section)

=== Cumulative distribution function ===

The [[cumulative distribution function]] can be expressed as:
: <math>F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i},</math>
where <math>\lfloor k\rfloor</math> is the "floor" under {{math|''k''}}, i.e. the [[floor and ceiling functions|greatest integer]] less than or equal to {{math|''k''}}.

It can also be represented in terms of the [[regularized incomplete beta function]], as follows:<ref>{{cite book |last=Wadsworth |first=G. P. |title=Introduction to Probability and Random Variables |year=1960 |publisher=McGraw-Hill |location=New York  |page=[https://archive.org/details/introductiontopr0000wads/page/52 52] |url=https://archive.org/details/introductiontopr0000wads |url-access=registration }}</ref>
: <math>\begin{align}
F(k;n,p) & = \Pr(X \le k) \\
&= I_{1-p}(n-k, k+1) \\
& = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt ,
\end{align}</math>
which is equivalent to the  [[cumulative distribution function]]s of the [[beta distribution]] and of the [[F-distribution|{{mvar|F}}-distribution]]:<ref>{{cite journal |last=Jowett |first=G. H. |year=1963 |title=The Relationship Between the Binomial and F Distributions |journal=Journal of the Royal Statistical Society, Series D |volume=13 |issue=1 |pages=55–57 |doi=10.2307/2986663 |jstor=2986663 }}</ref>
: <math>F(k;n,p) = F_{\text{beta-distribution}}\left(x=1-p;\alpha=n-k,\beta=k+1\right)</math>
: <math>F(k;n,p) = F_{F\text{-distribution}}\left(x=\frac{1-p}{p}\frac{k+1}{n-k};d_1=2(n-k),d_2=2(k+1)\right).</math>

Some closed-form bounds for the cumulative distribution function are given [[#Tail bounds|below]].