Lecture_20 - 1 TRANSFORMATION OF RANDOM VARIABLES(CONT Lecture 20 ORIE3500/5500 Summer2009 Chen Class Today • Transformation of Random

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Unformatted text preview: 1 TRANSFORMATION OF RANDOM VARIABLES (CONT.) Lecture 20 ORIE3500/5500 Summer2009 Chen Class Today • Transformation of Random Variables (PDF,PMF) • Transformation of Random Vectors 1 Transformation of Random Variables (Cont.) 1.1 PDF One uses similar approach in order to find the density of a transformed random variable, but one has to proceed with care. If Y = T ( X ), X is continuous and T is one-one, that is increasing or decreasing, then f Y ( y ) = f X ( T- 1 ( y )) fl fl fl dT- 1 ( y ) dy fl fl fl Example Suppose X is a standard exponential random variable and set Y = 1 /X . This means that X has pdf f X ( x ) = e- x ,x > . The transformation is T ( x ) = 1 /x and hence T- 1 ( y ) = 1 /y . This means that dT- 1 ( y ) dy =- 1 y 2 , and hence f Y ( y ) = f X (1 /y ) fl fl fl- 1 y 2 fl fl fl = 1 y 2 e- 1 /y ,y > . This method can be extended to the case when the transformation T is not one-one. Then the T ( x ) = y has several roots, T- 1 1 ( y ) ,T- 1 2 ( y ) ,... . If Y = T ( X ), then Y will have density X i f X ( T- 1 i ( y )) fl fl fl dT- 1 i ( y ) dy fl fl fl 1 1.2 PMF1 TRANSFORMATION OF RANDOM VARIABLES (CONT.) Example Let X be a standard normal random variable, and Y = T ( X ), where T ( x ) = ‰- x, x ≤ 2 x, x > . This means that the range of Y is [0 , ∞ ) and for y > T- 1 1 ( y ) =- y,T- 1 2 ( y ) = y/ 2 , for y = 0, T- 1 ( y ) =- y....
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell University (Engineering School).

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Lecture_20 - 1 TRANSFORMATION OF RANDOM VARIABLES(CONT Lecture 20 ORIE3500/5500 Summer2009 Chen Class Today • Transformation of Random

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