Problem 5 Suppose a student's learning progress and effort in a Statistics class, can be described by a scalar variable A. The instructor does not get to observe A directly, but assesses their course performance using two exams, whose scores we denote with X1 and X 2. The nal course grade is computed as a weighted average Y :2 le + (1 w)X2 for some choice of 11) between zero and one. Despite the instructor's best efforts, none of the assessments is perfect, and observed scores are X j = A + 65, for j = 1, 2, where 61, 62 are random variables representing other factors (\"luck\") affecting exam performance, which are unrelated to A. In particular, we assume that IE[61|A] = IE[62|A] = 0 and Var(61|A) = (If and Var(62|A) = 0%. (a) Compute the conditional expectation IE[Y|A]. Is the grading scheme \"fair\" for any choice of w? (b) Suppose that Cov(el, 2|A) = 0. Compute the conditional variance Var(Y|A) as a function of w. (c) Use your answer to (b) to nd the choice of 11) that minimizes Var(Y|A). If the instructor wishes to minimize the role of \"luck\