4) Let X, X,..., Xn be a random sample from a geometric distribution with parameter p....

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4) Let X₁, X₂,..., Xn be a random sample from a geometric distribution with parameter p. a. Find the Cramer-Rao lower bound for unbiased estimators of p. = 1 X₁ is a sufficient estimator for p. b. Show that T = c. Show that X belongs to a single parameter exponential family. (Hint: If x belongs to an exponential family then f(x; 0) = exp{t(0)T(x) + S(x) + (0)} in the usual notation). 4) Let X₁, X₂,..., Xn be a random sample from a geometric distribution with parameter p. a. Find the Cramer-Rao lower bound for unbiased estimators of p. = 1 X₁ is a sufficient estimator for p. b. Show that T = c. Show that X belongs to a single parameter exponential family. (Hint: If x belongs to an exponential family then f(x; 0) = exp{t(0)T(x) + S(x) + (0)} in the usual notation).

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