Probability

Second moment identity

The most-used rearrangement in these interviews, in both directions.

E[X2]=Var⁡(X)+(E[X])2\mathbb{E}[X^2] = \operatorname{Var}(X) + \big(\mathbb{E}[X]\big)^2

Where

Var⁡(X)\operatorname{Var}(X)
The variance, which is never negative.

Sanity check. It follows that E[X2]≥E[X]2\mathbb{E}[X^2] \ge \mathbb{E}[X]^2 always — a special case of Jensen.

Second moment identity · QuantMax