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Let $Y$ be a random variable and write $\mu = E[Y ]$. Show that $E[Y −\mu]=0$.


Let $X$ and $Y$ be random variables. Prove that $\text{Cov}[X, Y ] = E[X(Y − E[Y ])]$.

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Review your basic stats course. These are fundamental properties of the expected value. This is not economics. – Calvin Lin Jan 17 '13 at 2:59
This was a problem in my economics class... – Kevin L. Jan 18 '13 at 6:19

The covariance of $X$ and $Y$ is usually defined as $E\left((X-E(X))(Y-E(Y)\right)$. As suggested, we write $\mu$ for $E(Y)$. Note that $$(X-E(X))(Y-\mu)=X(Y-\mu) -(E(X))(Y-\mu).$$ Thus by the linearity of expectation, we have $$\text{Cov}(X,Y)=E(X(Y-\mu))-(E(X))E((Y-\mu) ).$$ But by the linearity of expectation, we have $$E(Y-\mu)=E(Y)-E(\mu)=\mu-\mu=0.$$

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