hw1_stat210a - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 Fall 2006 Issued: Thursday, August 31, 2006 Due: Thursday, September 7, 2006 Problem 1.1 Given a sequence of random variables such that Y n p μ , give one example where: (a) E ( Y n ) 6→ μ . (b) E ( Y n - μ ) 2 6→ 0. Problem 1.2 Suppose X i is a sequence of independent random variables with a common mean E ( X i ) = μ for all i and di±erent variances E ( X i - μ ) 2 = σ 2 i . (a) Show that ¯ X n = P n i =1 X i n L 2 μ as long as n i =1 σ 2 i = o ( n 2 ). Under the same assumptions, does ¯ X n p μ ? Why or why not? (b) Now consider the estimate e X n = P n i =1 1 σ i X i P n i =1 1 σ i . Prove that var( e X n ) var( ¯ X n ). Use this to conclude that, under the same conditions as in part (a), e X n p μ . Problem 1.3 Given a sequence of real-valued RVs such that Y n p μ , suppose moreover there exists M > 0 such that P ( | Y n | < M ) = 1 for all
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hw1_stat210a - UC Berkeley Department of Statistics STAT...

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