PROBLEM 0pts Reliable estimation of the mean u of a distribu

PROBLEM 0pts) Reliable estimation of the mean, u, of a distribution is by far, the most popular use of statistics However, an often eually important problem is estimation of the variance g When things go bad in a design, it is often the mean (or design) value that is messed up. However, when cheap components are used, it can easily happen that the mean is ok, but the variability becomes excessive. In this problem you will consider the situation where ur is known, but G is not. In this setting, the natural estimator, given X&) be iid with mean ur and variance oi,is 20e) (2.1) (a)(5pts) Use linearity of EC) to prove that EC 2 (e) o Proof (b) 5pts) Unlike PROBLEM 10b), it is not so simple to compute the variance, 2 (e) when Thas an arbitrary distribution. However, when has a normal pdf you have almost all of the tools to compute it directly. The only new tools you need are the following facts: Fact 1:Let Z Ihen which is, in words, a chi-squared pdf with 1 degree of freedom Fact 2: Let {W -1 iid random variables. Then g has a n pdf To utilize these facts, 20e) [Hint Z reformulate (2.1) so as to show that m Solution: (c)(5pts) The book does not give a very clear and simple description of the 20n) paf c f pp.287-289, if you wish.) For this reason it is better to got to https en.wikipedia.or wiki Chi-squared distribution. There, one finds that Au -n and or [which verifies (a), and (i ng 2g a 2m. Use this information and (b) to show that (i) u Solution: COMMENT: In summary, we find that if iid N then & a 2 /n (d)(5pts) Suppose that iidNO 50, g -3 Use the Matlab chi2cdf command to compute Pr

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