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Let X and Y be non-negative random variables, and let Z be another random variable, such that Z = max(X,Y). Prove that E[Z] ≤ E[X] + E[Y].
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This question can be proved by firstly assuming that E(Z) > E(X) + E(Y) holds under some conditions, then after a few derivations we can find that it will conflict with Z = max(X, Y).