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Unformatted text preview: STAT 410 Examples for 04/02/2008 Spring 2008 Central Limit Theorem X 1 , X 2 , , X n are i.i.d. with mean and variance 2 . ( ) X- n n = X 1 n n n i i & - = Z D , Z ~ N ( 0, 1 ). Theorem 4.3.9 ( ) & X- n n ( ) 2 , N D g ( x ) is differentiable t and g ' ( ) 0 ( ) ( ) ( ) X & g g n n- ( ) ( ) & 2 2 , & N ' g D 1. Let X 1 , X 2 , , X n be a random sample of size n from a Geometric ( p ) distribution ( the number of independent trials until the first success ) . That is, P ( X 1 = k ) = ( 1 p ) k 1 p , k = 1, 2, 3, . Recall that the maximum likelihood estimator of p , p , and the method of moments estimator of p , p ~ , are equal and p = p ~ = X 1 X 1 = = n i i n . a) Use WLLN ( Theorem 4.2.1 ) and Theorem 4.2.4 to show that p is a consistent estimator for p ( as n ) ....
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