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Unformatted text preview: a particular point (α, β ), we need to
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
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
fW,Z (α, β ) = 1
| 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
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
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.
- Spring '08
- The Land