# Probability, Expectation, Cov

• August 14th 2009, 04:28 AM
Maccaman
Probability, Expectation, Cov
Let X and Y be 2 random variables with $|\mathbb{E}[X]|, |\mathbb{E}[Y]|,$and $|\mathbb{E}[\frac{X}{Y}]|$ all finite, and with $\mathbb{P}(Y = 0) = 0$ and $\mathbb{E}[Y] \ne 0$.
Prove that $\mathbb{E}[\frac{X}{Y}] = \frac{\mathbb{E}[X]}{\mathbb{E}[Y]}$ if and only if Cov $(Y,\frac{X}{Y}) = 0$
• August 14th 2009, 04:35 AM
kobylkinks
solution
Use Cov(Y,X/Y)=E(Y*X/Y)-E(Y)*E(X/Y)=0. Because random variable Y*X/Y=X for all points except those where Y=0 the condition P(Y=0)=0
gives that E(X)=E(Y*X/Y).
• August 15th 2009, 01:01 AM
Maccaman
wow, that was surprisingly easy. Thanks for your help.