29. Stratified Sampling: Let Ui,..., U„ be independent random numbers and set = (i/f- + / - l)/n, i = 1, n. Hence, £/,·, / > 1, is uniform on

((/ - l)/n, i/n). J]?= é æÖ^/ç is called the stratified sampling estimator of

(a) Show that E[IU é = io*W

(b) Show that Var[E?= ! *(£/,)//!] < Var[E?e ! g(Ui)/n].

Hint: Let (7 be uniform (0, 1) and define Í by Í = i if

(/ - \)/n < U < i/n, i = 1 , n . Now use the conditional variance formula to obtain Varfe(£/)) = E[Yav(g(U)\N)] + Var(£[g(£/)|N])

>^[Var(g(C/)|^]

= £ Var(g(£/)|7V=/)_ £ Var^fi)]

/ = é ç i l = ç