Chapter 5, Problem 5.2TE

Show that $E\left[Y\right]={\displaystyle {\int }_0^{\infty }P\left\{Y>y\right\}dy}-{\displaystyle {\int }_0^{\infty }P\left\{Y<-y\right\}dy}$

Hint: Show that $\begin{array}{l}{\displaystyle {\int }_0^{\infty }P\left\{Y<-y\right\}dy=-{\displaystyle {\int }_{-\infty }^{0}x{f}_Y\left(x\right)dx}}\\ {\displaystyle {\int }_0^{\infty }P\left\{Y>y\right\}dy={\displaystyle {\int }_0^{\infty }x{f}_Y\left(x\right)dx}}\end{array}$