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] (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:

(a) Let X be a random variable with finite variance. Let Y = aX + 3 for some numbers a, B E R. Compute /1 = E [(Y - EY|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 1 X2. Compute 12 = E [(Y - E[Y|X])'] expressing your final result in terms of E[X*] for some value or values of k that you should specify. [3] (c) [TYPE:] Provide an interpretation of the quantities , and /2 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 I = E [(V - E[V U])2] into three parts as follows: - corr (U, V) - E[... ] * II III 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 = - Var(u) Cov(U,V) and start with / = E (V + 2U - ZU + E[V U])2]