100AHW6 - STAT 100A HWVI Due next Friday Problem 1 Suppose...

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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 0 = ( a - n/ 2) / ( n/ 2), and b 0 = ( b - n/ 2) / ( n/ 2). Show that P ( a X b ) R b 0 a 0 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 0 Z b 0 ). 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 . (2) Calculate E[ X ] and Var[ X ]. Problem 3: Consider the following joint probability mass function p ( x, y ) of the discrete random variables ( X, Y ): x \ y 1 2 3 1 .1 .1 .1 2 .2 .1 .2 3 .1 .05 .05 (1) Calculate p X ( x ) for x = 1 , 2 , 3. Calculate p Y ( y ) for y = 1 , 2 , 3.
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