SupposethatXandYarerandomvariablesandthatXandYarenonnegativeforallpointsinasamplespaceS.LetZbetherandomvariabledefinedbyZ(s)=max(X(s),Y(s))forallelementss∈S.showthatE(Z)≤E(X)+E(Y)E是期望函数~

Question

Suppose that X and Y are random variables and that X and Y are nonnegative for all points in a sample space S.Let Z be the random variable defined by Z(s)=max(X(s),Y(s)) for all elements s∈S.show that E(Z)≤E(X)+E(Y)
E是期望函数~

Answer 1

Proof:
Since X and Y are nonnegative,we have
Z(s)=max(X(s),Y(s))≤X(s)+Y(s).
that is
E(Z)≤E(X+Y)≤E(X)+E(Y).