Lecture_21 - 1 TRANSFORMATION OF RANDOM VARIABLES (CONT.)...

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Unformatted text preview: 1 TRANSFORMATION OF RANDOM VARIABLES (CONT.) Lecture 21 ORIE3500/5500 Summer2009 Chen Class Today Transformation of Random Vectors (Examples) Standard Transformations 1 Transformation of Random Variables (Cont.) Example This example will illustrate that we need to be careful with the range of Y 1 ,Y 2 sometimes. Suppose X 1 and X 2 are standard uniform random variables and let Y 1 = X 1 + X 2 ,Y 2 = X 1- X 2 . So the transform is ( y 1 ,y 2 ) = T ( x 1 ,x 2 ) = ( x 1 + x 2 ,x 1- x 2 ) , and so the inverse transform is x 1 = h 1 ( y 1 ,y 2 ) = 1 2 ( y 1 + y 2 ) ,x 2 = h 2 ( y 1 ,y 2 ) = 1 2 ( y 1- y 2 ) . Then we compute the Jacobian J T- 1 ( y 1 ,y 2 ) = fl fl fl fl 1 / 2 1 / 2 1 / 2- 1 / 2 fl fl fl fl =- 1 2 . The pdf of ( X 1 ,X 2 ) is f X 1 ,X 2 ( x 1 ,x 2 ) = 1 0 < x 1 ,x 2 < 1 0 otherwise = 1 [0 <x 1 ,x 2 < 1] . So the pdf of ( Y 1 ,Y 2 ) is f Y 1 ,Y 2 ( y 1 ,y 2 ) = f X 1 ,X 2 ( T- 1 ( y 1 ,y 2 )) fl fl fl J T- 1 ( y 1 ,y 2 ) fl fl fl = 1 2 1 [0 <y 1 + y 2 < 2 , <y 1- y 2 < 2] = 1 2 < y 1 + y 2 < 2 , < y 1- y 2 < 2 0 otherwise So one has to be careful with the range after transformation. 1 1 TRANSFORMATION OF RANDOM VARIABLES (CONT.) Example Suppose X 1 and X 2 are independent standard exponential random variables, and let Y 1 = X 1 + X 2 ,Y 2 = X 1 /X 2 . Find the joint pdf of....
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Lecture_21 - 1 TRANSFORMATION OF RANDOM VARIABLES (CONT.)...

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