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Unformatted text preview: ISTRIBUTED RANDOM VARIABLES Figure 4.26: (a) Mesh plots of both the standard bivariate normal, and the bivariate normal with
µX = 3, µY = −4, σX = 2, σY = 1, ρ = 0.5, shown on the same axes. (b) Contour plots of the same
pdfs. 4.11.2 Key properties of the bivariate normal distribution Proposition 4.11.2 Suppose X and Y have the bivariate normal pdf with paramters µX , µY , σX , σY ,
and ρ. Then
(a) X has the N (µX , σX ) distribution, and Y has the N (µY , σY ) distribution. (b) Any linear combination of the form aX + bY is a Gaussian random variable (i.e., X and Y
are jointly Gaussian).
(c) ρ is the correlation coeﬃcient between X and Y (i.e. ρX,Y = ρ).
(d) X and Y are independent if and only if ρ = 0.
(e) For estimation of Y from X , L∗ (X ) = g ∗ (X ). That is, the best unconstrained estimator g ∗ (X )
(f ) The conditional distribution of Y given X = u is N (L∗ (u), σe ), where σe is the MSE for
∗ (X ), given by (4.34) or (4.35).
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