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Unformatted text preview: < := Z A p ( x ) dx = P { X 1 A } , as . Dene, for each N , N = 1 N N i =1 1 { X i A } (1 B is the indicator function of set B ). (i) Find E ( N ) and Var( N ); (ii) Suppose N is large enough. For each > 0 and N > 0, using the attached Normal Table to determine z N such that the probability that N ( z N , + z N ) is (approximately) 0 . 99. (iii) Show that lim z N N = , a.s., no matter how large N is. 5. Let be a random variable with positive density function p ( x ). Suppose that p is twice dierentiable and satises the identity p ( x + y ) p ( x + y ) + p ( xy ) p ( xy ) = 2 p ( x ) p ( x ) , x,y ( , ) . Show that must be a normal random variable. 1...
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This document was uploaded on 01/25/2012.
 Spring '09
 Math

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