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Unformatted text preview: . Proposition 4.7.4 can also be viewed as the two dimensional generalization
of (3.6), where g is a function of one variable and g (g −1 (c)) plays the role that  det(J ) does in
Proposition 4.7.4.
Example 4.7.5 Let X , Y have the joint pdf:
fX,Y (u, v ) = u + v (u, v ) ∈ [0, 1]2
0
else . and let W = X 2 and Z = X (1 + Y ). Find the pdf, fW,Z . 148 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Solution: The vector (X, Y ) in the u − v plane is transformed into the vector (W, Z ) in the α − β
plane under a mapping g that maps u, v to α = u2 and β = u(1 + v ). The image in the α − β plane
of the square [0, 1]2 in the u − v plane is the set A given by
√
√
A = {(α, β ) : 0 ≤ α ≤ 1, and α ≤ β ≤ 2 α}.
See Figure 4.20. The mapping from the square is onetoone, becasue if (α, β ) ∈ A then (u, v ) can
! 2 A v
1 1 " u
1 Figure 4.20: Transformation from the u − v plane to the x − y plane.
be recovered by u = √ α and v = β
√
α − 1. The determinant of the Jacobian is det(J ) = det 2u
0
1+v u = 2u2 = 2α. Therefore, P...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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