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Unformatted text preview: rameters µX , µY , σX , σY , ρ satisfy σX > 0, σY > 0 and −1 < ρ < 1. As shown
below, µX and µY are the means of X and Y , respectively, σX and σY are the standard deviations,
respectively, and ρ is the correlation coeﬃcient. We shall describe some properties of the bivariate
normal pdf in the remainder of this section.
There is a simple way to recognize whether a pdf is a bivariate normal. Namely, such a pdf has
the form: fX,Y (u, v ) = C exp(−P (u, v )), where P is a second order polynomial of two variables:
P (u, v ) = au2 + buv + cv 2 + du + ev + f. 4.11. JOINT GAUSSIAN DISTRIBUTION 169 The constant C is selected so that the pdf integrates to one. Such a constant exists if and only if
P (u, v ) → +∞ as |u| + |v | → +∞, which requires a > 0, c > 0 and b2 − 4ac < 0. Without loss of
generality, we can take f = 0, because it can be incorporated into the constant C. Thus, the set of
bivariate normal pdfs can be parameterized by the ﬁve parameters: a, b, c, d, e. 4.11.1 From the standard 2-d n...
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