Question: 17. Let X be a nonnegative random variable with distribution function F. Define I (t) = B 1 if X > t 0 otherwise. (a)

17. Let X be a nonnegative random variable with distribution function F. Define I (t) = B 1 if X > t 0 otherwise.

(a) Prove that # ∞ 0 I (t) dt = X.

(b) By calculating the expected value of both sides of part (a), prove that E(X) = E ∞ 0 4 1 − F (t)5 dt. This is a special case of Theorem 6.2.

(c) For r > 0, use part

(b) to prove that E(Xr ) = r E ∞ 0 t r−14 1 − F (t)5 dt.

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