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Unformatted text preview: STAT 100A HWVI Due next Friday Problem 1: Suppose we flip a fair coin n times independently. Let X be the number of heads. Let k = n/ 2 + z √ n/ 2, or z = ( k n/ 2) / ( √ n/ 2). Let g ( z ) = P ( X = k ). (1) Using the Stirling formula n ! ∼ √ 2 πnn n e n , show that g (0) ∼ 1 √ 2 π 2 √ n . a ∼ b means that a/b → 1 as n → ∞ . (2) Show that g ( z ) /g (0) → e z 2 / 2 as n → ∞ . (3) For two integers a < b , let a = ( a n/ 2) / ( √ n/ 2), and b = ( b n/ 2) / ( √ n/ 2). Show that P ( a ≤ X ≤ b ) → R b a f ( z ) dz , where f ( z ) = 1 √ 2 π e z 2 / 2 . (4) Let Z = ( X n/ 2) / ( √ n/ 2). Show that P ( a ≤ X ≤ b ) = P ( a ≤ Z ≤ b ). Argue that in the limit Z ∼ N(0 , 1). Problem 2: Suppose Z ∼ N(0 , 1). The density of z is f ( z ) = 1 √ 2 π e z 2 / 2 . E[ Z ] = 0, Var[ Z ] = 1. Let X = μ + σZ , where σ > 0. (1) Find the probability density function of X ....
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This note was uploaded on 01/27/2011 for the course STATISTICS 100a taught by Professor Cristou during the Winter '10 term at UCLA.
 Winter '10
 CRISTOU

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