Let X and Y be independent, positive r.v.s. with finite expected values.

(a) Give an example where , computing both sides exactly.*Hint: start by thinking about the simplest examples you can think of!*

(b) If X and Y are i.i.d., then is it necessarily true that ?

(c) Now let X ~ Gamma(a, ) and Y ~ Gamma(b, ). Show without using calculus that for every real c > 0.

(a) Give an example where , computing both sides exactly.

(b) If X and Y are i.i.d., then is it necessarily true that ?

(c) Now let X ~ Gamma(a, ) and Y ~ Gamma(b, ). Show without using calculus that for every real c > 0.

Solution:
(a) Numerous examples possible. (b) Yes, by symmetry and linearity. (c) As shown in class in the bank-post office story, X/(X + Y) is independent of X + Y . So Xc/(X + Y)c is independent of (X + Y)c, which shows that they are uncorrelated.

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