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Let X1, X2,...,Xn be independent random variables from Normal (mu, sigma_i^2), i=1, 2,...,n, where mu is...

Let X1, X2,...,Xn be independent random variables from Normal (mu, sigma_i^2), i=1, 2,...,n, where mu is unknown and sigma_1^2, sigma_2^2,...,sigma_n^2 are known. Determine a sufficient statistic for mu. Derive the UMVUE and MLE of T(mu)=mu^3.

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