29. Stratified Sampling: Let U1, . . . ,Un be independent random numbers and set

¯U i = (Ui + i − 1), i=1, . . . , n. Hence, ¯Ui , i1, is uniform on ((i − 1), i). ni

=1 g(

¯U i) is called the stratified sampling estimator of

1 0 g(x) dx.

(a) Show that E[ni

=1 g(

¯U i)] =

1 0 g(x) dx.

(b) Show that Var[ni

=1 g(

¯U i)] Var[ni

=1 g(Ui)].

Hint: Let U be uniform (0, 1) and define N by N = i if (i − 1)

Transcribed Image Text:

0, b 1, b g(x) 0,0 1,0