In each case below, determine whether the estimator under consideration is unbiased, asymptotically unbiased, and/or consistent.

a. The random sample \(\left(X_{1}, \ldots X_{n}ight)\) is generated from a Gamma population distribution. The estimator \(t(\mathbf{X})=\sum_{i=1}^{n} X_{i} / n\) will be used to estimate \(\mathrm{E}\left(X_{i}ight)=\alpha \beta\).

b. The random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from an exponential population distribution. The estimator \(t(\mathbf{X})=(1 / 2) \sum_{i=1}^{n} X_{i}^{2} / n\) will be used to estimate \(\operatorname{var}\left(X_{i}ight)=\theta^{2}\).

c. The random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from a geometric population distribution. The estimator \(t(\mathbf{X})=\left(S^{2}-\bar{X}ight)\) will be used to estimate \(E\left(X_{i}^{2}ight)=p^{-2}\).

d. The random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from a Bernoulli population distribution. The estimator \(t(\mathbf{X})=\bar{X}(1-\bar{X})\) will be used to estimate \(\operatorname{var}\left(X_{i}ight)=\) \(p(1-p)\). (Hint: \(\left.2 \sum_{i=1}^{n} \sum_{j>i}^{n} a=n(n-1) aight)\).