1 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18

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: 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 2 2 (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 ) is linear. 2 2 (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). L 2 2...
View Full Document

Ask a homework question - tutors are online