We say that X isstochastically smallerthan Y (X ≤stY ) if P{X ≤ t} ≥ P{Y ≤ t} for all t. Show that if X and Y have finite expectations and X≤stY , then E(X) ≤ E(Y ).

(Hint: Start with nonnegative X and Y and use Theorem 7.2.2. Then use the decomposition into a positive and negative part.) Show also that the converse assertion is false: there exist random variables X and Y such that E(X) < E(Y ) and X is not stochastically smaller than Y .