Isye 2027

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

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

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

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