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

a. The iid random sample ð Þ X1; ... ; Xn is generated from a Gamma population distribution. The estimatortð Þ X

¼ X will be used to estimate the value of ab.

b. The iid random sample ð Þ X1; ... ; Xn is generated from an exponential population distribution. The estimator tð Þ¼ X ð Þ n  1 1 Pn i¼1 Xi  X  2 will be used to estimate the value of y2.

c. The iid random sample ð Þ X1; ... ; Xn is generated from a Poisson population distribution. The estimatortð Þ X

¼ Xn will be used to estimate the value of l.

d. The iid random sample ð Þ X1; ... ; Xn is generated from a Poisson population distribution. The estimator tð Þ¼ X ð Þ n  1 1 Pn i¼1 Xi  X  2 will be used to estimate the value of l.

e. The iid random sample ð Þ X1; ... ; Xn is generated from a geometric population distribution. The estimator tð Þ¼ X S2 n will be used to estimate 1ð Þ  p p2.

f. The iid random sample ð Þ X1; ... ; Xn is generated from a Bernoulli population distribution. The estimator tðXÞ ¼ X 1  X   will be used to estimate pð Þ 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) ¼ X/n will be used to estimate p.