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# Hw 4 - Stat 116 Homework 4 Solutions 1 The Beta density is...

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Stat 116 Homework 4 Solutions April 23, 2008 1. The Beta( α, β ) density is f ( x ) = x α - 1 (1 - x ) β - 1 B ( α,β ) which is proportional to g ( x ) = x α - 1 (1 - x ) β - 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 ) = ( α - 1) x α - 2 (1 - x ) β - 1 - ( β - 1)(1 - x ) β - 2 x α - 1 . Setting this equal to zero, we get: ( α - 1) x α - 2 (1 - x ) β - 1 = ( β - 1)(1 - x ) β - 2 x α - 1 , which implies that ( α - 1)(1 - x ) = ( β - 1) x , from which we get x = α - 1 α + β - 2 . (a) α > 1 , β > 1: Now α > 1 means that g (0) = 0, and β > 1 means g (1) = 0, so the maximum density cannot be at x = 0 or x = 1, so it must occur at x = α - 1 α + β - 2 . (b) This condition translates to α 1 , β 1 and either α or β or both is strictly less than one. First suppose α < 1 , β < 1. Now α < 1 means that g (0) = , and β < 1 means g (1) = , so that means x = 0 and x = 1 are the maxima, and x = α - 1 α + β - 2 is the minimum, so the density must be U- shaped. Alternatively α = 1 , β < 1 means that g ( x ) = (1 - x ) β - 1 , so g ( x ) = (1 - β )(1 - x ) β - 2 > 0, so g

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Hw 4 - Stat 116 Homework 4 Solutions 1 The Beta density is...

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