Let X be a random variable with finite variance. Let Y = X + for some numbers , R. Compute l1 = E [(Y E[Y|X])2]. [3] (b) Again, let X be a random variable with finite variance. Additionally, let A U(1, 1) such that X and A are independent and consider Y = AX. You may use without proof that A2 X2. Compute l2 = E [(Y E[Y|X])2] expressing your final result in terms of E[Xk] for some value or values of k that you should specify. [3] (c) [TYPE:] Provide an interpretation of the quantities l1 and l2 obtained in parts (a) and (b) above in terms of the ability to predict Y given the value of X and explain any difference you observe. [3]

(a) Let X be a random variable with finite variance. Let Y = aX + [3 for some numbers a, 6 R. Compute l1 2 E [(Y IE[Y|X])2]. [3] (b) Again, let X be a random variable with finite variance. Additionally, let A N U(1, 1) such that X and A are independent and consider Y = AX. You may use without proof that A2 J. X2. Compute [2 = E [(Y lE[Y|X])2] expressing your final result in terms of IE1[X'\"] for some value or values of k that you should specify. [3] (c) [TYPE:] Provide an interpretation of the quantities 21 and Z2 obtained in parts (a) and (b) above in terms of the ability to predict Y given the value of X and explain any difference you observe. [3] (d) Consider two random variables U and V with mean zero and variance one. Derive a formula that decomposes l = ]E [(V E[V|U])2] into three parts as follows: l=\\1/,COFI'(U,V)Ell (*) I II 111 You need to fill in the missing part in the expectation on the right hand side. [TYPE:] Provide a statistical interpretation of each of the three terms on the right hand side of (*) I, II and III, commenting on the case when U and V are (i) independent, (ii) linearly related as in part (a) above and (iii) nonlinearly related with corr(U, V) = 0. [6] Hint: Let z = 9% and start with l 2 E [(V + 2U zU + 1E[V|U])2]