10. Let X1, ..., Xn be i.i.d. from some distribution Qo, and let X = (X1+ ... + Xn)/n be the sample average. (a) Show that S² = E(X; – X)²/(n – 1) is unbiased for o? = = Varø(X;). | (b) If Qo is the Bernoulli distribution with success probability 0, then a unique best unbiased estimator for o? can be obtained by calculating E[ô²|X], where ở² is some unbiased estimator. Find this unique best unbiased estimator. (c) If Qo is the exponential distribution with failure rate 0, compute an unbiased esti- mator ô? for o² = 1/0² in terms of X. (d) Again, consider the scenario given in part (c), where the calculation of the unique best unbiased estimator using E[ô²\X] also applies. Give a formula for E[X?|X = c].

10. Let X1, ..., Xn be i.i.d. from some distribution Qo, and let X = (X1+ ... + Xn)/n be the sample average. (a) Show that S² = E(X; – X)²/(n – 1) is unbiased for o? = = Varø(X;). | (b) If Qo is the Bernoulli distribution with success probability 0, then a unique best unbiased estimator for o? can be obtained by calculating E[ô²|X], where ở² is some unbiased estimator. Find this unique best unbiased estimator. (c) If Qo is the exponential distribution with failure rate 0, compute an unbiased esti- mator ô? for o² = 1/0² in terms of X. (d) Again, consider the scenario given in part (c), where the calculation of the unique best unbiased estimator using E[ô²\X] also applies. Give a formula for E[X?|X = c].

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Estimation

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10. Let X1,..., Xn be i.i.d. from some distribution Qe, and let X = (X1 + ... + Xn)/n be the
sample average.
(a) Show that S² = E(X; – X)²/(n – 1) is unbiased for o? = Varo(X;).
(b) If Qe is the Bernoulli distribution with success probability 0, then a unique best
unbiased estimator for o? can be obtained by calculating E[ô²|X], where ở² is some
unbiased estimator. Find this unique best unbiased estimator.
(c) If Qe is the exponential distribution with failure rate 0, compute an unbiased esti-
mator ô? for o² = 1/0² in terms of X.
(d) Again, consider the scenario given in part (c), where the calculation of the unique
best unbiased estimator using E[ô²|X] also applies. Give a formula for E[X?|X = c].

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