stat210a_HW01 - UC Berkeley Department of Statistics STAT...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 - Solutions Fall 2009 Issued: Tuesday, September 1, 2009 Due: Tuesday, September 8, 2009 Problem 1.1 1. Y n = ± 0 , with probability 1 - 1 n n, with probability 1 n . For all δ > 0, 0 P ( | Y n - 0 | ≥ δ ) 1 n . Hence, for all δ > 0, lim n →∞ P ( | Y n - 0 | ≥ δ ) = 0, which implies that Y n p 0. However, E ( Y n ) = 1 for all n and thus lim n →∞ E ( Y n ) = 1 9 0. 2. Let Y n = - n, with probability 1 2 n 0 , with probability 1 - 1 n n, with probability 1 2 n . Similarly as in part (a), for all δ > 0, 0 P ( | Y n - 0 | ≥ δ ) 1 n and thus lim n →∞ P ( | Y n - 0 | ≥ δ ) = 0, so Y n p 0. Also we have μ = E ( Y n ) = 0. On the other hand, E ( Y n - μ ) 2 = 2( n ) 2 1 2 n = 1 for all n, so lim n →∞ E ( Y n - μ ) 2 = 1 9 0. Problem 1.2 (a) It is enough to prove that lim n →∞ E | Y n - μ | = 0. Notice that for any fixed ± > 0: 0 E | Y n - μ | = Z | Y n - μ | 1 ( | Y n - μ | ≤ ± )d P ( Y n ) + Z | Y n - μ | 1 ( | Y n - μ | > ± )d P ( Y n ) ± · 1 + s Z | Y n - μ | 2 d P ( Y n ) Z 1 ( | Y n - μ | > ± )d P ( Y n ) ± + p ( M + | μ | ) 2 · P ( | Y n - μ | > ± ) . Let n → ∞ , the second term above converges to 0, due to convergence in probability. It follows that for any ± > 0, 0 lim n →∞ E | Y n - μ | ≤ ± Hence, lim n →∞ E | Y n - μ | = 0, which proves the conclusion. (b) The same example in Problem 1.1 part(b) disproves the claim. (c) For any ± > 0 and positive interger k , we have: 0 E | Y n - μ | k = Z | Y n - μ | k I ( | Y n - μ | ≤ ± ) dP ( Y n ) + Z | Y n - μ | k I ( | Y n - μ | > ± ) dP ( Y n ) ± k + ( | M | + | μ | ) k · P ( | Y
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stat210a_HW01 - UC Berkeley Department of Statistics STAT...

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