Chapter Summary

Chapter Summary

Key Points

• 1.

  The multivariate Gaussian $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ is fully specified by its mean and covariance. The PDF involves the precision matrix $\boldsymbol{\Sigma}^{-1}$ in the exponent, and the constant-density contours are ellipsoids whose axes align with the eigenvectors of $\boldsymbol{\Sigma}$.

• 2.

  Marginals are Gaussian. If $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ is partitioned into $(\mathbf{X}_1, \mathbf{X}_2)$, then $\mathbf{X}_1 \sim \mathcal{N}(\boldsymbol{\mu}_1, \boldsymbol{\Sigma}_{11})$ — simply read off the corresponding block.

• 3.

  Conditionals are Gaussian with Schur complement formulas. $\mathbf{X}_1 | \mathbf{X}_2 = \mathbf{x}_2 \sim \mathcal{N}(\boldsymbol{\mu}_{1|2}, \boldsymbol{\Sigma}_{1|2})$ where $\boldsymbol{\mu}_{1|2} = \boldsymbol{\mu}_1 + \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}(\mathbf{x}_2 - \boldsymbol{\mu}_2)$ is affine in $\mathbf{x}_2$ and $\boldsymbol{\Sigma}_{1|2} = \boldsymbol{\Sigma}_{11} - \boldsymbol{\Sigma}_{12}\boldsymbol{\Sigma}_{22}^{-1}\boldsymbol{\Sigma}_{21}$ does not depend on $\mathbf{x}_2$.

• 4.

  Affine transformations preserve Gaussianity. $\mathbf{A}\mathbf{X} + \mathbf{b} \sim \mathcal{N}(\mathbf{A}\boldsymbol{\mu} + \mathbf{b}, \mathbf{A}\boldsymbol{\Sigma}\mathbf{A}^T)$. Every linear combination of Gaussian components is scalar Gaussian.

• 5.

  Uncorrelated $\Longleftrightarrow$ independent for Gaussians. This is the defining structural advantage of the Gaussian family: second-order analysis (decorrelation, PCA, whitening) achieves full statistical independence.

• 6.

  The whitening transform maps $\mathbf{X}$ to $\mathbf{W} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$. It is the multivariate analogue of standardization and is the first step in many detection and estimation algorithms.

• 7.

  Chi-squared and Wishart distributions arise from quadratic functions of Gaussians. The Mahalanobis distance $(\mathbf{X} - \boldsymbol{\mu})^T\boldsymbol{\Sigma}^{-1}(\mathbf{X} - \boldsymbol{\mu}) \sim \chi^2_n$. The sample covariance matrix follows a Wishart distribution.

• 8.

  The proper complex Gaussian $\mathcal{CN}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ models baseband noise and Rayleigh fading. Circular symmetry (vanishing pseudo-covariance) means the distribution is invariant to phase rotation. All real Gaussian properties carry over.