Let X have the pdf f X (k ) ( 1) k1 k , k 1, 2, 3, , 1, which is geometric (p 1 ) For this pdf E(X) and Var(X) ( 1) Is the statistic X efficient

Question: Let X have the pdf f X (k; ) = ( 1) k1 / k , k = 1, 2, 3, . . .,

Let X have the pdf f_X(k; θ) = (θ − 1)^k−1/θ^k, k = 1, 2, 3, . . ., θ >1, which is geometric (p = 1/θ). For this pdf E(X) = θ and Var(X) = θ(θ −1). Is the statistic X efficient?

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