Question: 5. Use the result that for a nonnegative random variable Y, $$E[Y] = int_{0}^{infty} P(Y > t) dt$$ to show that for a nonnegative random

5. Use the result that for a nonnegative random variable Y,

$$E[Y] = \int_{0}^{\infty} P(Y > t) dt$$

to show that for a nonnegative random variable X,

$$E[X^n] = \int_{0}^{\infty} nx^{n-1} P(X > x) dx$$

HINT: Start with

$$E[X^n] = \int_{0}^{\infty} P(X^n > t) dt$$

and make the change of variables r = xn.

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