48 moments of jointly distributed random variables

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Unformatted text preview: a particular point (α, β ), we need to Z Y Z consider values of (u, v ) such that g (u, v ) = (α, β ). The simplest case is if there is at most one value of (u, v ) so that g (u, v ) = (α, β ). If that is the case for all (α, β ), g is said to be a one-to-one function. If g is one-to-one, then g −1 (α, β ) is well-deﬁned for all (α, β ) in the range of g. These observations lead to the following proposition: Proposition 4.7.4 Suppose W = g ( X ), where X has pdf fX,Y , and g is a one-to-one mapping Z Y Y from the support of fX,Y to R2 . Suppose the Jacobian J of g exists, is continuous, and has nonzero determinant everywhere. Then W has joint pdf given by Z fW,Z (α, β ) = 1 fX,Y | det J | g −1 α β . for (α, β ) in the support of fW,Z . Proposition 4.7.4 generalizes Proposition 4.7.1. Indeed, suppose A is a two-by-two matrix with det(A) = 0 and let g be the linear mapping deﬁned by g (u, v ) = A u . Then g satisﬁes the v bypotheses of Proposition 4.7.4, the Jacobian function of g is constant and equal to A everywhere, and g −1 ( α ) = A−1 α . Therefore, the conclusion of Proposition 4.7.4 reduces to the conclusion β β of Proposition 4.7.1...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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