Here’s a result that I’ve found to be very useful, but have re-derived over and over and over again.

Consider a general mixed central moment,

$\mathrm{E}((X_1-\mathrm{E}(X_1)) \dotsi (X_m-\mathrm{E}(X_m)))$

Moments of this form can be expressed as one of $m$ different covariances,

$\mathrm{Cov}((X_1-\mathrm{E}(X_1)) \dotsi (X_{j-1}-\mathrm{E}(X_{j-1})) \dotsi (X_{j+1}-\mathrm{E}(X_{j+1})) \dotsi (X_m-\mathrm{E}(X_m)), X_j)$

It always seems kind of obvious, but having had my intuition wrong many times before I usually get a little suspicious — which causes me to re-derive it to make sure I’m not messing anything up. Never again. The proof is actually very simple…just expand one of the deviations. Without loss of generality, this will look simpler if we set $j = m$,

$\mathrm{E}((X_1-\mathrm{E}(X_1)) \dotsi (X_{m-1}-\mathrm{E}(X_{m-1})) (X_m-\mathrm{E}(X_m)))$
$\mathrm{E}((X_1-\mathrm{E}(X_1)) \dotsi (X_{m-1}-\mathrm{E}(X_{m-1})) X_m - (X_1-\mathrm{E}(X_1)) \dotsi (X_{m-1}-\mathrm{E}(X_{m-1})) \mathrm{E}(X_m))$

And by the linearity of expected value,

$\mathrm{E}((X_1-\mathrm{E}(X_1)) \dotsi (X_{m-1}-\mathrm{E}(X_{m-1})) X_m) - \mathrm{E}((X_1-\mathrm{E}(X_1)) \dotsi (X_{m-1}-\mathrm{E}(X_{m-1}))) \mathrm{E}(X_m)$

This expression is in a common form for the covariance,

$\mathrm{Cov}(x, y) = \mathrm{E}(xy) - \mathrm{E}(x)\mathrm{E}(y)$

which proves the result.