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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.
 Spring '07
 Staff
 Probability

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