Lecture 8 — Gaussian Random Vectors
🎓 Probability — Lecture 8

Gaussian Random Vectors

A random vector $X \in \mathbb{R}^d$ is Gaussian if every linear form $a^\top X$ is univariate normal. We write $X \sim \mathcal{N}(\mu, \Sigma)$ with mean $\mu$ and covariance $\Sigma \succeq 0$.

Definition & Density

Density (full-rank $\Sigma$):

$$f_X(x) = \frac{1}{(2\pi)^{d/2} \, (\det\Sigma)^{1/2}} \exp\!\Big(-\tfrac12 (x-\mu)^\top \Sigma^{-1} (x-\mu)\Big).$$

  • $\mu = \mathbb{E}[X]$, $\Sigma = \operatorname{Cov}(X) = \mathbb{E}[(X-\mu)(X-\mu)^\top]$.
  • Characteristic function: $\varphi_X(t) = \exp\{ i t^\top \mu - \tfrac12 t^\top \Sigma t\}$.
  • Affine invariance: If $Y = A X + b$, then $Y \sim \mathcal{N}(A\mu + b, A\Sigma A^\top)$.

If $\Sigma$ is singular, density lives on a lower-dimensional affine subspace; the characteristic function still characterizes the law.

Interactive 2D Gaussian Sandbox