Let be independent random variables, each with unknown mean and unknown variance . Further, let be independent random

variables (and independent of ), also with mean and variance

(a) Show that is an unbiased estimator of , where and

are the respective sample means of the two samples.

(b) Show that

(c) Show that is minimized when

(d) Show that

is an unbiased estimator of , where and are the respective sample variances of the two samples.