Fall 2011 Homework 9: Question 1
Let X and Y be independent, positive r.v.s. with finite expected values.
(a) Give an example where $E(\frac{X}{X+Y})\neq \frac{E(X)}{E(X+Y)}$, 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 $E(\frac{X}{X+Y})= \frac{E(X)}{E(X+Y)}$?
(c) Now let X ~ Gamma(a, $\lambda$) and Y ~ Gamma(b, $\lambda$). Show without using calculus that $E(\frac{X^{c}}{(X+Y)^{c}})=\frac{E(X^{c})}{E((X+Y)^{c})}$ 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.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."