(4) Let X1,,Xn be a random sample from a Binomial(1,) population, where (0,1). We know that the MLE of is i=1nXi/n. Thus, if we want to estimate 2=(1), the variance of the Binomial (1,) population, a reasonable estimator would be ^2:=ni=1nXi(1ni=1nXi). If we denote X:=i=1nXi, then XBin(n,) and ^2=nX(1nX) 1 (a) Show that ^2 is a biased estimator of 2. (b) Is ^2 an asymptotically unbiased estimator of 2 (see Definition 6.5.3 in the notes)?.

1. (4) Let X1,,Xn be a random sample from a Binomial(1,) population, where (0,1). We know that the
MLE of is i=1nXi/n. Thus, if we want to estimate 2=(1), the variance of the Binomial (1,) population,
a reasonable estimator would be ^2:=ni=1nXi(1ni=1nXi). If we denote X:=i=1nXi, then XBin(n,) and
^2=nX(1nX) 1 (a) Show that ^2 is a biased estimator of 2. (b) Is ^2 an asymptotically unbiased
estimator of 2 (see Definition 6.5.3 in the notes)?