Chapter 8: Multivariate Distributions and the Gaussian

Learning Objectives

• Define the multivariate Gaussian distribution and write its PDF in terms of the precision matrix
• Derive the marginal and conditional distributions of a partitioned Gaussian vector using Schur complements
• Prove that affine transformations of Gaussian vectors remain Gaussian
• Establish that uncorrelated Gaussian components are independent — a property unique to the Gaussian family
• Perform the whitening transform and interpret its geometric meaning
• Connect chi-squared, Wishart, and Student-t distributions to Gaussian random vectors
• Interpret the eigendecomposition of the covariance matrix as a rotation to principal axes
• Derive the moment generating function and characteristic function of the multivariate Gaussian
• Define the proper complex Gaussian distribution and explain circular symmetry
• Apply multivariate Gaussian theory to LMMSE estimation and Rayleigh fading models