Hw 4 - Stat 116 Homework 4 Solutions April 23, 2008 1. The...

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Unformatted text preview: Stat 116 Homework 4 Solutions April 23, 2008 1. The Beta( , ) density is f ( x ) = x - 1 (1- x ) - 1 B ( , ) which is proportional to g ( x ) = x - 1 (1- x ) - 1 , and thus has the same extrema. Since beta densities by construction are smooth, the potential extrema are x = 0, x = 1, and the roots of the derivative g ( x ) = ( - 1) x - 2 (1- x ) - 1- ( - 1)(1- x ) - 2 x - 1 . Setting this equal to zero, we get: ( - 1) x - 2 (1- x ) - 1 = ( - 1)(1- x ) - 2 x - 1 , which implies that ( - 1)(1- x ) = ( - 1) x , from which we get x = - 1 + - 2 . (a) > 1 , > 1: Now > 1 means that g (0) = 0, and > 1 means g (1) = 0, so the maximum density cannot be at x = 0 or x = 1, so it must occur at x = - 1 + - 2 . (b) This condition translates to 1 , 1 and either or or both is strictly less than one. First suppose < 1 , < 1. Now < 1 means that g (0) = , and < 1 means g (1) = , so that means x = 0 and...
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This note was uploaded on 01/13/2010 for the course STATS 116 taught by Professor Staff during the Spring '07 term at Stanford.

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Hw 4 - Stat 116 Homework 4 Solutions April 23, 2008 1. The...

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