Question: 19. Let X be a nonnegative random variable with distribution function F. Define (a) Prove that R 0 I(t) dt = X. (b) By

19. Let X be a nonnegative random variable with distribution function F. Define

I(t)= if X > t otherwise.

(a) Prove that R ∞
0 I(t) dt = X.

(b) By calculating the expected value of both sides of part (a), prove that

image text in transcribed

This is a special case of Theorem 6.2.

(c) For r > 0, use part

(b) to prove that

image text in transcribed

I(t)= if X > t otherwise.

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