= Fourth centered moment of a binomial distribution
{title2=$\mathbb E(W-np)^4=3n^2q^2+nq(1-6q),\quad q=p(1-p)$}
For $W\sim\operatorname{Bin}(n,p)$, expand the fourth power of the sum of centered independent Bernoulli variables. Terms with an index occurring once vanish; single-index fourth moments contribute $nq(1-3q)$ and paired second moments contribute $3n(n-1)q^2$. Their sum is the displayed formula. Scaling by $n^4$ yields a uniform $O(n^{-2})$ fourth moment for $W/n-p$.
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