Chapter 5, Problem 5.5TE

Use the result that for a nonnegative random variable $E\left[Y\right]={\displaystyle {\int }_0^{\infty }P\left\{Y>t\right\}dt}$ to show that for a nonnegative random variable $E\left[X^n\right]={\displaystyle {\int }_0^{\infty }n{x}^{n-1}P\left\{X>x\right\}dx}$.

Hint: Start with $E\left[X^n\right]={\displaystyle {\int }_0^{\infty }P\left\{X^n>t\right\}dt}$ and make the change of variables $t=x^n$.