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

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

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