Stat 5101 Lecture Notes - School of Statistics

80 Stat 5101 (Geyer) Course Notes2-30. Verify the moments of the U(a, b) distribution given in Section B.2.1 ofAppendix B.2-31. The proof of Corollary 2.35 used cor(X, X) = 1 without comment. Provethis.2-32. Suppose X ∼ Gam(α, λ).(a)For any real number p>−α, the p-th ordinary momentα p = E(X p )exists. Calculate it.Hint: Follow the pattern of Example 2.5.3. Your answer will involvegamma functions that cannot be simplified using the recurrence relation ifp is not an integer (which we didn’t say it was).(b)Show thatvar(X) = α λ 2Hint: Use part (a) and the recurrence relation for gamma functions, (B.3)in Appendix B.2-33. Suppose X has probability density(note the domain).f(x) = 3 x 4, x > 1(a) For what positive integers k is X k in L 1 ?(b)Calculate E(X k ) for the positive integers k such that the expectation exists.2-34. Suppose X has probability density(note the domain).f(x) = 12 √ x , 0