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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1- Solutions Fall 2006 Issued: Thursday, August 31, 2006 Due: Thursday, September 7, 2006 Problem 1.1 Solution to 1. Let: Y n = , with probability 1- 1 n n, with probability 1 n Clearly, E ( Y n ) = 1 for all n and, hence, lim n E ( Y n ) = 1. However, for all > 0, P ( | Y n- | ) 1 n and hence, for all > 0, lim n P ( | Y n- | ) = 0 so Y n p 0. 2. Let: Y n = - n, with probability 1 2 n , with probability 1- 1 n n, with probability 1 2 n Hence E ( Y n ) = 0 and var ( Y n ) = 2 ( n ) 2 1 2 n = 1 for all n and, therefore, lim n var ( Y n ) = 1. As in item a, for all > 0, 0 P ( | Y n- | ) 1 n and hence, for all > 0, lim n P ( | Y n- | ) = 0 so Y n p 0. Problem 1.2 See Examples from section 2.2 in Large Sample Theory, by Erich Lehmann: 1. We have that E ( X- ) 2 = E h P n i =1 ( X i- ) 2 n 2 i . Given independence, E ( X- ) 2 = P n i =1 2 i n 2 0 establishing convergence in quadratic mean ( L 2-convergence). Convergence in probability follows from convergence in quadratic mean. 1 2. Once it is proved that the var ( X n ) var ( X n ), L 2 convergence of X n implies L 2 convergence of X n . Convergence in probability follows. The statement is true in both the original form X = P i X i i P i 1 i and the corrected form X = P i X i 2 i P i 1 2 i For the corrected form: Write the mean as the estimate for in the regression model X i = +...
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