Editing Binomial distribution (section)

=== Higher moments ===
<!-- Please stop changing the equation \mu_1 = 0, it is correct. The first central moment is not the mean. -->
The first 6 [[central moment]]s, defined as <math> \mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right] </math>, are given by 
: <math>\begin{align}
\mu_1 &= 0, \\ 
\mu_2 &= np(1-p),\\
\mu_3 &= np(1-p)(1-2p),\\
\mu_4 &= np(1-p)(1+(3n-6)p(1-p)),\\
\mu_5 &= np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\
\mu_6 &= np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^2 p^2 (1-p)^2).
\end{align}</math>

The non-central moments satisfy
: <math>\begin{align}
\operatorname {E}[X] &= np, \\ 
\operatorname {E}[X^2] &= np(1-p)+n^2p^2,
\end{align}</math>
and in general
<ref name="Andreas2008">
{{citation 
 |last1=Knoblauch |first1=Andreas 
 |title=Closed-Form Expressions for the Moments of the Binomial Probability Distribution 
 |year=2008
 |journal=SIAM Journal on Applied Mathematics 
 |url=https://www.jstor.org/stable/40233780
 |volume=69 
 |issue=1 
 |pages=197–204
 |doi=10.1137/070700024 
 |jstor=40233780 
|url-access=subscription 
 }}</ref><ref name="Nguyen2021">
{{citation 
 |last1=Nguyen |first1=Duy 
 |title=A probabilistic approach to the moments of binomial random variables and application 
 |year=2021
 |journal=The American Statistician 
 |url=https://www.tandfonline.com/doi/abs/10.1080/00031305.2019.1679257?journalCode=utas20
 |volume=75
 |issue=1 
 |pages=101–103 
 |doi=10.1080/00031305.2019.1679257 
 |s2cid=209923008 
|url-access=subscription 
 }}</ref>
: <math>
\operatorname {E}[X^c] = \sum_{k=0}^c \left\{ {c \atop k} \right\} n^{\underline{k}} p^k,
</math>
where <math>\textstyle \left\{{c\atop k}\right\}</math> are the [[Stirling numbers of the second kind]], and <math>n^{\underline{k}} = n(n-1)\cdots(n-k+1)</math> is the <math>k</math>th [[Falling and rising factorials|falling power]] of <math>n</math>.
A simple bound
<ref>{{Citation |last1=D. Ahle |first1=Thomas |title=Sharp and Simple Bounds for the raw Moments of the Binomial and Poisson Distributions
|year=2022
|volume=182
|doi=10.1016/j.spl.2021.109306
|journal=Statistics & Probability Letters
|page=109306 |arxiv=2103.17027 }}</ref> follows by bounding the Binomial moments via the [[Poisson distribution#Higher moments|higher Poisson moments]]: 
: <math>
\operatorname {E}[X^c] \le
\left(\frac{c}{\ln(c/(np)+1)}\right)^c \le (np)^c \exp\left(\frac{c^2}{2np}\right).
</math>
This shows that if <math>c=O(\sqrt{np})</math>, then <math>\operatorname {E}[X^c]</math> is at most a constant factor away from <math>\operatorname {E}[X]^c</math>