Contours of Marginal PDFs for Gaussian Random Variables

Contours of Marginal PDFs for Gaussian Random Variables 

Two methods are introduced in this appendix for drawing the contours of marginal PDFs. Method 1. Eigenvalue Problem Method 

Consider a vector of two Gaussian random variables 0 = [6, 62V with mean 0 = [0J, l and covariance matrix E. The goal here is to obtain the parametric form of the joint PDF contour that covers an area with a prescribed probability. First, define a vector of two new random variables y = [yi, y2] by the following transformation  

If the matrix PH P is diagonal, say D, the Gaussian random variables y and y2 are uncorrelated and, hence, statistically independent. Then, it is an easy task to draw the PDF contours in the y —9 — y2 coordinate system. In order to obtain a solution for the matrix P to fulfill this goal, consider the eigenvalue problem of the covariance matrix  

Since the covariance matrix Z is symmetric, the eigenvector matrix V can be normalized such that V = V. In particular, this matrix takes the following form  

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