L14_S10 - AMS 311 Fall Semester, 2010 Chapter Six: Jointly...

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AMS 311 Fall Semester, 2010 Chapter Six: Jointly Distributed Random Variables 6.7. Joint Probability Distribution of Functions of Random Variables Transformation of two random variables is a crucial problem and hard to handle. It is important to review your multivariable calculus so that you are up to speed technically. The probability theory is given in this section and may look hard. The calculations are straightforward but require careful attention. Let 1 X and 2 X be jointly continuous random variables with joint probability density function ). , ( 2 1 , 2 1 x x f X X In the discussion below, I will refer to 1 X as 1 old and 2 X as . 2 old Let ) , ( 2 1 1 1 X X g Y = and ). , ( 2 1 2 2 X X g Y = I will refer to 1 Y as 1 new and 2 Y as . 2 new Assume that the functions ) , ( 2 1 1 1 x x g y = and ) , ( 2 1 2 2 x x g y = can be uniquely solved for 1 x and 2 x . Further assume that 0 ) det( ) , ( 2 2 1 2 2 1 1 1 2 1 = x g x g x g x g x x J at all points
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L14_S10 - AMS 311 Fall Semester, 2010 Chapter Six: Jointly...

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