Prerequisites & Notation

Before You Begin

This chapter brings together ideas from several earlier chapters. We need joint distributions (Chapter 7), covariance and expectation (Chapter 5–6), and basic linear algebra (eigendecomposition, positive semi-definiteness). If any item below is unfamiliar, revisit the linked material first.

Joint PDFs, marginal and conditional distributions(Review ch07)
Self-check: Given a joint PDF $f_{X,Y}(x,y)$ , can you compute $f_{X|Y}(x|y)$ ?
Covariance, variance, and the correlation coefficient(Review ch07)
Self-check: Can you compute $\text{Cov}(X,Y)$ from a joint distribution?
Eigenvalue decomposition of symmetric matrices
Self-check: Given a $2 \times 2$ symmetric matrix, can you find its eigenvalues and eigenvectors?
Positive semi-definite matrices
Self-check: Can you verify that $\mathbf{x}^T \mathbf{A} \mathbf{x} \geq 0$ for all $\mathbf{x}$ ?
The scalar Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$ (Review ch06)
Self-check: Can you write the PDF of $\mathcal{N}(\mu, \sigma^2)$ and compute $Q(x)$ ?
Matrix multiplication, transpose, inverse, and determinant
Self-check: Can you compute $(\mathbf{A}\mathbf{B})^T$ and $\det(\mathbf{A})$ for $2 \times 2$ matrices?

Notation for This Chapter

Symbols introduced or heavily used in this chapter. Bold lowercase denotes vectors, bold uppercase denotes matrices.

Symbol	Meaning	Introduced
$\mathbf{X} = (X_1, \ldots, X_n)^T$	Random vector	s01
$\boldsymbol{\mu} = \mathbb{E}[\mathbf{X}]$	Mean vector	s01
$\boldsymbol{\Sigma}$	Covariance matrix $\mathbb{E}[(\mathbf{X} - \boldsymbol{\mu})(\mathbf{X} - \boldsymbol{\mu})^T]$	s01
$\mathbf{R}$	Correlation matrix $\mathbb{E}[\mathbf{X}\mathbf{X}^T]$	s01
$\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$	Multivariate Gaussian distribution	s02
$\boldsymbol{\Sigma}^{-1}$	Precision (information) matrix	s02
$\mathcal{CN}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$	Proper complex Gaussian distribution	s08
$\chi^2_n$	Chi-squared distribution with $n$ degrees of freedom	s07
$Q(x)$	Gaussian tail probability	s00

← Ch 7 Random Vectors and Their Statistics