Show that the covariance is also $\mathrm{cov}(X, Y) = E[(X-E(X))Y] = E[X(Y-E(Y))]$

Question

The covariance between $X$ and $Y$ is given by $cov(X, Y) = E[(X-E(X))(Y-E(Y))]$. However, I was told that an equivalent expression for covariance is $$\mathrm{cov}(X, Y) = E[(X-E(X))Y] = E[X(Y-E(Y))]$$ Why is this true? Isn't the expressions only true for when $E(Y) = 0$ or $E(X) = 0$?

Answer 1

Note that $E(E(X)(Y-E(Y))=E(X)E(Y-E(Y))=0$

Hence, $E(X(Y-E(Y))=E(X(Y-E(Y))+E(X)E((Y-E(Y))=E(X(Y-E(Y))+E(E(X)(Y-E(Y))$

And use the linearity of the expectation.