27. Let X l 9 X n be independent random variables with E\X^\ = 0, Vai(XÏ) = ó2, / = É , . , . , Ë , and consider estimates of 0 of the form

Ó?= é kiXi where £*= ë ë,· = 1. Show that Var(£?= x ë^) is minimized when

ë, = (1/ïÀ)/(Ó"=éà<Þ),À= 1. . . . . Ë.

Possible Hint: If you cannot do this for general n, try it first when

ç = 2.

The following three problems are concerned with the estimation of ioS(x)dx = E[g(U)] where U is uniform (0, 1).