# Question

Since the variances of the mean and the midrange are not affected if the same constant is added to each observation, we can determine these variances for random samples of size 3 from the uniform population

By referring instead to the uniform population

(a) Show that E(X) = 1/2 , E(X2) = 1/3 , and var(X) = 1/12 for this population so that for a random sample of size n = 3, var(X) = 1/36 .

(b) Use the results of Exercises 8.46 and 8.52 on page 254 (or derive the necessary densities and joint density) to show that for a random sample of size n = 3 from this population, the order statistics Y1 and Y3 have E(Y1) = 14 , E(Y21) = 1/10 , E(Y3) = 34 , E(Y23) = 35 , and E(Y1Y3) = 1/5 so that var(Y1) = 3/80 , var(Y3) = 3/80, and cov(Y1,Y3) = 1/80.

(c) Use the results of part (b) and Theorem 4.14 on page 135 to show that E Y1 + Y3 2 = 1 2 and var(Y1 + Y3 / 2) = 1 40 and hence that for random samples of size n = 3 from the given uniform population, the midrange is unbiased and more efficient than the mean.

By referring instead to the uniform population

(a) Show that E(X) = 1/2 , E(X2) = 1/3 , and var(X) = 1/12 for this population so that for a random sample of size n = 3, var(X) = 1/36 .

(b) Use the results of Exercises 8.46 and 8.52 on page 254 (or derive the necessary densities and joint density) to show that for a random sample of size n = 3 from this population, the order statistics Y1 and Y3 have E(Y1) = 14 , E(Y21) = 1/10 , E(Y3) = 34 , E(Y23) = 35 , and E(Y1Y3) = 1/5 so that var(Y1) = 3/80 , var(Y3) = 3/80, and cov(Y1,Y3) = 1/80.

(c) Use the results of part (b) and Theorem 4.14 on page 135 to show that E Y1 + Y3 2 = 1 2 and var(Y1 + Y3 / 2) = 1 40 and hence that for random samples of size n = 3 from the given uniform population, the midrange is unbiased and more efficient than the mean.

## Answer to relevant Questions

Show that if Θ is a biased estimator of θ, then Substituting “asymptotically unbiased” for “ unbiased” in Theorem 10.3, show that X + 1 / n+ 2 is a consistent estimator of the binomial parameter θ. After referring to Example 10.4, is the nth order statistic, Yn, a sufficient estimator of the parameter β? Given a random sample of size n from a continuous uniform population, use the method of moments to find formulas for estimating the parameters α and β. Given a random sample of size n from a Pareto population (see Exercise 6.21 on page 184), use the method of maximum likelihood to find a formula for estimating its parameter α.Post your question

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