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Unformatted text preview: 21 . The half-peak level set is π the set of (α, β ) pairs such that fW,Z (α, β ) is equal to half of the peak value, or {(α, β ) : α2 + β 2 = 2 ln 2 ≈ (1.18)2 }, which is a circle centered at the origin with radius approximately 1.18. Since all the bivariate normal pdfs are obtained from fW,Z by linear scaling and translation, the half-peak level set of a general bivariate normal pdf completely determines the pdf. The half-peak level set for the general bivariate normal fX,Y given by (4.37) is 2 2 u−µX v −µY u−µX v −µY + σY − 2ρ σX σX σY 2 (u, v ) : = 2 ln 2 ≈ (1.18) , 1 − ρ2 which is an ellipse. The space of ellipses in the plane is five dimensional–two coordinates specify the center of an ellipse, two numbers give the lengths of the major and minor axes, and a final number gives the angle of the major axis from the horizontal. This gives another way to parameterize the set of bivariate normal pdfs using five parameters. 170 CHAPTER 4. JOINTLY D...
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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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