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

a. The iid random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from a Gamma population distribution. The estimator \(t(\mathbf{X})\) \(=\bar{X}\) will be used to estimate the value of \(\alpha \beta\).

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

c. The iid random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from a Poisson population distribution. The estimator \(t(\mathbf{X})\) \(=\bar{X}_{n}\) will be used to estimate the value of \(\lambda\).

d. The iid random sample \(\left(X_{1}, \ldots, X_{n}ight)\) is generated from a Poisson population distribution. The estimator \(t(\mathbf{X})=(n-1)^{-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}ight)^{2}\) will be used to estimate the value of \(\lambda\).

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

f. The iid 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 \(p(1-p)\).

g. The random sample of size \(1, X\), is generated from a Binomial population distribution for which the value of the parameter \(n\) is known. The estimator \(t(X)=X / n\) will be used to estimate \(p\).