Let X be a nonnegative r.v. with a continuous, strictly increasing CDF F, and let =

QUESTION:

Let X be a nonnegative r.v. with a continuous, strictly increasing CDF F, and let = E(X). The previous problem asks for a proof that Z 1 0 F 1 (u)du = . In this problem, you can assume this result. The goal is to understand the following identity: E(X) = Z 1 0 P(X>x)dx. This result is the continuous analog of Theorem 4.4.8. (a) Give a visual explanation of the identity for E(X) by drawing a picture of a CDF and interpreting a certain area in two dierent ways. (b) Explain why we can write X = Z 1 0 I(X t)dt, where in general if Yt is an r.v. for each t 0, we define R 1 0 Ytdt to be the r.v. whose value is R 1 0 Yt(s)dt when the outcome of the experiment is s. Assuming that swapping an E and an integral is allowed (which can be done using results from real analysis), derive the identity for E(X).