In this problem, we prove the identity var(X) = E[var(X|Y )] + var(E[X|Y ]). Assume X and Y are zero mean. You may use that var(X|Y ) = E[(X E[X|Y ])2 |Y ].

(a) First, show that E[var(X|Y )] = var(X E[X|Y ]).

(b) Draw X, E[X|Y ], and X E[X|Y ] in the Hilbert space of random variables. Specify and justify the angle between E[X|Y ] and X E[X|Y ].

(c) Use the previous two parts to conclude the identity var(X) = E[var(X|Y )] + var(E[X|Y ]).