NotesOct20

# 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 ) dT - 1 ( y ) dy .
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 ) dT - 1 i ( y ) dy .
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 discrete), or a known joint pdf (if continuous). Let ( Y 1 , . . . , Y k ) = T ( X 1 , . . . , X k ) be a function, or a transformation, of ( X 1 , . . . , X k ). Find the joint pmf of ( Y 1 , . . . , Y k ) in the discrete case; find the joint pdf of ( Y 1 , . . . , Y k ) in the con- tinuous case.
If ( X 1 , . . . , X k ) is discrete, then ( Y 1 , . . . , Y k ) = T ( X 1 , . . . , X k ) is also discrete. The joint pmf of Y 1 , . . . , Y k ): p Y 1 ,...,Y k ( y j 1 , . . . , y j k ) = X x i 1 ,...,x

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• '10
• SAMORODNITSKY
• Probability theory, yk

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