Xn be a random sample from Let X,, X2, Beta (a, B) where a is known- Show that is sufficient for a. I- X;

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Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o2 is unknown. Consider the estimate n 1 Σ(X₁-1 for the variance. a) Show that 7² C := i=1 -μ)² 72 2 ·Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x2 (n) by qn(a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o² = 4 vs. H₁: o² #4 at significance level 5%.

We have the following information about the random variables X and Y: 0.5, дY – 1,o = 1, o = 2.25. Calculate the variance of Z 1X + 7Y, | a) when the coefficient of correlation is p(X, Y) = 0.33: b) when X and Y are independent random variables: