Problem 7.53 (a) Let X1,...Xn be a random sample taken from an exponential distribu[1]tion with parameter . Show that X is an unbiased estimator of

Problem 7.53

(a) Let X1,...Xn be a random sample taken from an exponential distribu[1]tion with parameter λ. Show that X is an unbiased estimator of 1/λ.

(b) Let U = min(X1,...,Xn). Show that U is exponentially distributed with parameter nλ. Hint: Show that P(U>x) = exp(−nλx).

(c) Deduce from

(a) that nU is an unbiased estimator of 1/λ.

(d) Compute V (nU) and V (X). Which estimator is better? Justify your answer by stating an appropriate criterion.