2. (a) Let X be a random variable with finite variance. Let Y = aX + 3 for some numbers o, 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])2] 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 l, and 12 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: 1 = - corr (U, V) - E .. . ] (* ) 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 = - Cov(U,V) Var(U) and start with / = E (V + 2U - ZU + EV|U])"]