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Unformatted text preview: UC Berkeley, Department of Statistics Fall, 2009 STAT 210A: Theoretical Statistics HW#1 Due: In class, September 08, 2009 Problems related to appendix materials 1.1. Given a sequence of random variables such that P n Y μ → , give one example where: (a) ( ) n E Y μ . (b) 2 ( ) n E Y μ- 0. 1.2. Given a sequence of real-valued RVs such that P n Y μ → , suppose moreover there exists M > such that 2 ( ) n E Y M < for all n . Prove or disprove the following claims: (a) ( ) n E Y → μ . (b) 2 ( ) n E Y μ- → 0. If (| | ) 1 n P Y M < = for all n , prove or disprove (c) | | k n E Y μ- → 0 for any positive integer k . 1.3. Suppose i X is a sequence of independent random variables with a common mean ( ) i E X μ = for all i and different variances 2 2 ( ) i i E X μ σ- = . (a) Show that 1 2 ( ) / n n i i L X X n μ = = → ∑ if 2 2 1 ( ) n i i o n σ = = ∑ . Under the same assumptions, • does 1 ( ) / P n n i i X X n μ = = → ∑ ? Why or why not?...
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at Berkeley.
- Fall '08