In contrast the correlation coecient xy is

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 one-to-one, 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...
View Full Document

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.

Ask a homework question - tutors are online