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# hw3 - AMS 553 Homework 3 1(L&K 8.7 For a < b the...

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AMS 553: Homework 3 a < b , the right-triangular distribution has density function f R ( x ) = ( 2( x - a ) ( b - a ) 2 if a x b 0 otherwise. (1) and the left-triangular distribution has density function f L ( x ) = ( 2( b - x ) ( b - a ) 2 if a x b 0 otherwise. (2) These distributions are denoted by RT ( a,b ) and LT ( a,b ), respectively. (a) Show that if X RT (0 , 1), then X 0 = a + ( b - a ) X RT ( a,b ); verify the same relation between LT (0 , 1) and LT ( a,b ). Thus it is suﬃcient to generate from RT (0 , 1) and LT (0 , 1). (b) Show that if X RT (0 , 1), then 1 - X LT (0 , 1). Thus it is enough to restrict our attention further to generating from RT (0 , 1). (c) Derive the inverse-transform algorithm for generating from RT (0 , 1). Despite the result in ( b ), also derive the inverse-transform algorithm for generating directly from LT (0 , 1). (d) As an alternative to the inverse-transform method, show that if
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• Spring '08