Simple mathematical relationships like $V(X) = E(X^2) - E(X)^2$, aside from being theoretical results, are useful because they allow analysts to do back-of-the-envelope calculations, restate results for simpler interpretation and visualization, and come up with new statistics on the fly.

I'm asking for a useful expansion or identity, and an example of its usefulness. Obviously there are many; keep it to one identity per answer so that they can be voted on individually.

The identities don't have to be as "simple" as the variance expansion above, but they should be accessible to applied researchers who perhaps don't have training in formal mathematical stats.

They also don't need to be popular or standard; part of the point of this question is to ferret out some more obscure but useful tricks that one might not encounter in an intro stats or regression class.

This is related to What theories should every statistician know? but with the scope constrained to specific mathematical expressions.

Simple mathematical relationships like $V(X) = E(X^2) - E(X)^2$, aside from being theoretical results, are useful because they allow analysts to do back-of-the-envelope calculations, restate results for simpler interpretation and visualization, and come up with new statistics on the fly.

I'm asking for a useful expansion or identity, and an example of its usefulness. Obviously there are many; keep it to one identity per answer so that they can be voted on individually.

The identities don't have to be as "simple" as the variance expansion above, but they should be accessible to applied researchers who perhaps don't have training in formal mathematical stats.

They also don't need to be popular or standard; part of the point of this question is to ferret out some more obscure but useful tricks that one might not encounter in an intro stats or regression class.

This is related to What theories should every statistician know? but with the scope constrained to specific mathematical expressions.