Chapter 5, Problem 5.3TE

Show that if X has density function f. then $E\left[g\left(X\right)\right]={\displaystyle {\int }_{-\infty }^{\infty }g\left(x\right)f\left(x\right)dx}$

Hint: Using Theoretical Exercise 5.2, start with $E\left[g\left(X\right)\right]={\displaystyle {\int }_0^{\infty }P\left\{g\left(x\right)>y\right\}dy}-{\displaystyle {\int }_0^{\infty }P\left\{g\left(x\right)<-y\right\}dy}$ and then proceed as in the proof given in the text when $g\left(x\right)>0$.