7. We say that X is stochastically larger than Y, written X Y, if for all 1,

$$P(X> r) \ge P(Y > 1)$$

Show that if X Y, then E[X] [Y] when

(a) X and Y are nonnegative random variables;

(b) X and Y are arbitrary random variables.

HINT: Write X as

$$X = X^+ - X^-$$

where

$$X^+ = \begin{cases}

X &\text{if } X \ge 0\\

0 &\text{if } X < 0

\end{cases}$$

$$X^- = \begin{cases}

0 &\text{if } X \ge 0\\

-X &\text{if } X < 0

\end{cases}$$

Similarly, represent Y as YY. Then make use of part (a).