Your course grade depends on two test scores: X1 and X2. Your score X on test i is Gaussian (μ = 74, σ = 16) random variable, independent of any other test score.
(a) With equal weighting, grades are determined by Y = X1/2 + X2/2. You earn an A if Y > 90. What is P[A] = P(Y > 90]?
(b) A student asks the professor to choose a weight factor w, 0 < W < 1, such that
Y wX1 + (1 - w)X2.
Find P[A] as a function of the weight w. What value or values of w maximize P[A] = P[Y > 90}?
(c) A different student proposes that the better exam is the one that should count and that grades should be based on M = max(X1, X2). In a fit of generosity, the professor agrees Now what is P[A] = P[M > 90]?
(d) How generous was the professor? In a class of 100 students, what is the expected increase in the number of A's awarded?