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Suppose that X1, X2, ..., Xn are independent random variables (not iid) with densities ÍXi(z10,) -.2 e _ θ:/z1(z > 0), where θί 〉 0, for i = 1, 2, , n. (a) Derive the form of the likelihood ratio test (LRT) statistic for testing versuS H1: not Ho. You do not have to find the distribution of the likelihood ratio test (LRT) statistic under Ho- Just find the form of the statistic. (b) From your result in part (a), deduce that TL for any positive T1,T2, ..., Tn (c) Suppose Ho is true so that the common distribution of X1, X2, ..., Xn, now viewed as being conditional on θ, is described by where θ > 0. Identify a conjugate prior for θ. » Specify any hyperparameters in your prior (pick values for fun if you want). Show how to carry out the (Bayesian) test of H0 : θ-1 versus H1 : θ 1.

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