Assume i.i.d. samples X1, . . . , Xn from pdf f(x). We take the sample average Ib1 = 1 n Pn i=1 h(Xi) as an approximate to E[h(X)] for some function h. Alternatively, we can sample Y1, . . . , Yn i.i.d. from another pdf g(x), and form the sample average Ib2 = 1 n Pn i=1 h(Yi) f(Yi) g(Yi) . 1. (10 points.) Show that E[Ib1] = E[Ib2] = E[h(X)], i.e., they are both unbiased. 2. (10 points.) Show that if we choose g(x) = |h(x)|f(x) R u |h(u)|f(u)du, E[Ib2 2 ] E[Ib2 1 ]. (Hint: use CauchySchwarz inequality)