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Unformatted text preview: − ln(A), which can be shown to have he Rayleigh distribution
with parameter σ, and let Θ = 2πB, which is uniformly distributed on [0, 2π ]. Then let X =
R cos(Θ) and Y = R sin(Θ). 4.7.3 Transformation of pdfs under a manytoone mapping In the previous subsection we considered onetoone mappings g . If a mapping g from R2 to R2
has a continuous Jacobian with a nonzero determinant, but there are multiple points in the u − v
plane mapping to a single point in the α − β plane, then Proposition 4.7.4 requires a modiﬁcation
in order to apply. Namely, the pdf of fW,Z (α, β ) is given by a sum of terms, with the sum running
over points of the form (ui , vi ) such that g (ui , vi ) = (α, β ). This modiﬁcation is illustrated by an
example for which the mapping is twotoone.
Example 4.7.7 Suppose W = min{X, Y } and Z = max{X, Y }, where X and Y are jointly
continuoustype random variables. Express fW,Z in terms of fX,Y . 150 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES v "
U
u ! Figure 4.21: The mapping of (u, v ) to (min{u, v }, max{u, v }).
Solution: Note tha...
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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
 Zahrn
 The Land

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