The expectation of the maximum of RVs is greater or equal to the maximum of the expectations

For any sequence of RVs X1,…,Xn the following holds:

E[max{X1, …, Xn}] >= max{E[X1], …, E[Xn]}    (*)

Proof. We have that max{X1, …, Xn} >= Xi for all i. Thus, this holds in expectation as well, i.e. E[max{X1, …, Xn}] >= E[Xi] for all i. Since this holds for all i, it holds for their maximum as well.

Interpretation: “the expected best idea out of a fixed population of ideas dominates the expected value of any individual idea (even the best one!)”. Variability plays a role here.

When does (*) hold with equality and when with strict inequality?

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