If X is a random variable having the binomial distribution with the parameters n and θ, show that n ∙ X/n ∙ (1 – X/n) is a biased estimator of the variance of X.
Answer to relevant QuestionsIf a random sample of size n is taken without replacement from the finite population that consists of the positive integers 1, 2, . . . , k, show that (a) The sampling distribution of the nth order statistic, Yn, is given ...Show that for the unbiased estimator of Example 10.4, n + 1 / n ∙ Yn, the Cramer-Rao inequality is not satisfied. If X1, X2, and X3 constitute a random sample of size n = 3 from a normal population with the mean µ and the variance σ2, find the efficiency of X1 + 2X2 + X3 / 4 relative to X1 + X2 + X3 / 3 as estimates of µ. If Θ1 = X/n , Θ2 = X + 1 / n+ 2 , and Θ3 = 1/3 are estimators of the parameter θ of a binomial population and θ = 1/2 , for what values of n is (a) The mean square error of Θ2 less than the variance of Θ1; (b) The ...If X1 and X2 are independent random variables having binomial distributions with the parameters θ and n1 and θ and n2, show that X1 + X2 / n1 + n2 is a sufficient estimator of θ.
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