NotesOct20 - Computing the pdf of Y = T (X ). Let X be...

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Computing the pdf of Y = T ( X ) . Let X be continuous with pdf f X . Sometimes Y = T ( X ) is also continuous. To compute the density of Y adjust by the derivative of the inverse transfor- mation .
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Monotone transformations : the function T is either increasing or decreasing on the range of X . A monotone function T is automatically one- to-one. The pdf of Y : f Y ( y ) = f X ± T - 1 ( y ) ² ³ ³ ³ ³ ³ dT - 1 ( y ) dy ³ ³ ³ ³ ³ .
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Example Let X be a standard exponential random variable. Compute the density of Y = 1 /X .
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This method sometimes works even when the function T is not monotone. The equation T ( x ) = y may have several roots, T - 1 1 ( y ), T - 1 2 ( y ) ,... . Each of the roots contributes to the den- sity of Y = T ( X ): f Y ( y ) = X i f X ± T - 1 i ( y ) ² ³ ³ ³ ³ ³ ³ dT - 1 i ( y ) dy ³ ³ ³ ³ ³ ³ .
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Example Let X be standard normal, and Y = T ( X ), with T ( x ) = ( - x if x 0 2 x if x > 0. Compute the pdf of Y .
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Transformations of random vectors Statement of the problem: let ( X 1 ,...,X k ) be a random vector with a known joint pmf (if
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This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).

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NotesOct20 - Computing the pdf of Y = T (X ). Let X be...

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