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# 1. Demonstrate that the unbiased ("n -1") estimator of variance is unbiased, that the plug-in ("n") estimator of variance is biased, and that both

1. Demonstrate that the unbiased ("n -1") estimator of variance is unbiased, that the plug-in ("n") estimator of variance is biased, and that both the "n-1" and the "n" estimators of standard deviation are biased as follows. First, simulate 1,000,000 samples of n = 5 observations per sample from the N(70,102 ) distribution. (The true variance is σ2 = 100 and the true standard deviation is σ = 10.) Calculate the 1,000,000 estimates of σ2 using the "n - 1" formula and the 1,000,000 estimates of σ2 using the plug-in "n" formula, then take the square roots f these to obtain the estimates of the standard deviation. The bias of the estimator is the expected value of the estimator minus the estimand. Here, the estimands are known. The expected value of the estimator can be approximated, by the Law of Large numbers, as the average of a large number of replicates of the estimate. Here, there are 1,000,000 replicates, a large number. A. Estimate the bias of "n -1" variance estimator. (Note: since this estimator is in fact unbiased, any difference between the bias estimate and 0 is explained by chance alone). B. Estimate the bias of "n" variance estimator. (Note: This estimator really is biased.) C. Estimate the bias of "n -1" standard deviation estimator. (Note: The estimand is different here than in A.) D. How does Jensen's inequality apply to the answer in C.? (Hint: See the first paragraph of p. 294) E. Estimate the bias of "n " standard deviation estimator. (Note: The estimand is different here than in A.)