2. (a) Let X be a random variable with finite variance. Let Y = ozX + it? for some numbers (1,6 6 ER. Compute t1 = E [(Y w E{Y|X])2]. [3] (b) Again, let X be a random variable with finite variance. Additionally. let A ~ (C) (d) U(1, 1) such that X and A are independent and consider Y = AX. You may use without proof that A2 i X 2. Compute I2 = ]E[(Y 1E[Y|X])2] expressing your final result in terms of ]E[X"] for some value or values of k that you should specify. [3] [TYPE] Provide an interpretation of the quantities 1 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] Consider two random variables U and V with mean zero and variance one. Derive a formula that decomposes i = ]E [(V lE[V|U])2] into three parts as follows: l=+(corr(:',V)) lEI[u-] (*) 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 (a). 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 oorr(U, V) = 0. [6] Hint: Let 2 = 3033'; and start with l = ]E [(v + zU zU ]E[V|U])2]. Note: A previous version of this part question was missing the square for the correlation term