information:extraction
Higher moments

The first 6 central moments, defined as μ c = E) c ] {\displaystyle \mu _{c}=\operatorname {E} \left)^{c}\right]}, are given by

The non-central moments satisfy

and in general

where { c k } {\displaystyle \textstyle \left\{{c \atop k}\right\}} are the Stirling numbers of the second kind, and n k _ = n (n − 1) ⋯ (n − k + 1) {\displaystyle n^{\underline {k}}=n(n-1)\cdots (n-k+1)} is the k {\displaystyle k} th falling power of n {\displaystyle n}.A simple bound follows by bounding the Binomial moments via the higher Poisson moments:

This shows that if c = O (n p) {\displaystyle c=O({\sqrt {np}})}, then E {\displaystyle \operatorname {E} } is at most a constant factor away from E c {\displaystyle \operatorname {E} ^{c}}