(a) Let Z ~ N(0, 1) and let Y be a r.v. such that P(Y = V3) = P(Y = -V3) = 1/6; P(Y = 0) = 2/3. Show that E(Z*) = E(Y*) for k = 1, 2,3,4. (Hint: use the MGF to compute the moments of Z.) (b) Consider a random variable Y ~ Unif(01,62), 61 0, 8 > 0, which has density f(@) = r(@)T(B) Let a > 0 be a constant. Show that E(Yo) = !(a to)f(a +8) r(or(a+B+a) (Hint: this uses the common proof technique of getting the integrand to look like a density so it integrates to one.) 2 (d) The SAT and ACT are two standardized college entrance exams that produce scores that are approximately Normally distributed with means 480 for the SAT and 18 for the ACT, and standard deviations 100 for the SAT and 6 for the ACT. One school sets 485 as the minimum SAT score required for admission. What minimum ACT score would allow admission for an equal proportion of students? (You may use a normal table or a normal probability calculator)