Let [math]X,Y[/math] be any two random variables with a joint density function. Suppose that

[[math]]\operatorname{E}[X|Y] = g(Y), \, g(y) = E[X|Y=y].[[/math]]

Which of the following statements is always true:

• [math]|\operatorname{Cov}(X,Y)| \lt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
• [math]|\operatorname{Cov}(X,Y)| \gt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
• [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math]
• If [math]\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0[/math] then [math]X [/math] and [math]Y[/math] are independent.
• If [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math] for every [math]Y[/math] then [math]X = \operatorname{E}[X | Y][/math].