Question: Assume \(Y=1+2X+u\), where X, Y, and u are r.v.’s, u is independent of X with E(u)=0 and \(\operatorname{var}\left( u \right)={{\sigma }^{2}}\).

a) Calculate \(E\left( u|X=0 \right)\), \(E\left( u|X=3 \right)\), \(E\left( Y|X=0 \right)\), \(E\left( Y|X=3 \right)\), \(E\left( u|X \right)\), and \(E\left( Y|X \right)\)

b) Calculate \(\operatorname{var}\left( u|X=0 \right)\), \(\operatorname{var}\left( u|X=3 \right)\), \(\operatorname{var}\left( u|X \right)\), \(\operatorname{var}\left( Y|X=0 \right)\), \(\operatorname{var}\left( Y|X=3 \right)\), and \(\operatorname{var}\left( Y|X \right)\)