Question: Let X be a positive continuous r.v. with p.d.f. fX and c.d.f. FX. (a) Show that a necessary condition for E [X] to exist is

 Let X be a positive continuous r.v. with p.d.f. fX and c.d.f. FX.

(a) Show that a necessary condition for E [X] to exist is limx→∞ x [1 − FX (x)]

= 0. Use this to show that the expected value of a Cauchy random variable does not exist.

(b) Prove via integration by parts that E [X] =  ∞

0 [1 − FX (x)] dx if E [X] < ∞.

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