Multivariate Normal — Section 12: Multivariate Statistics

The multivariate Normal $N(\mu, \Sigma)$ is the workhorse of high-dimensional statistics. Its density is

f(x) = \frac{1}{(2\pi)^{k/2} |\Sigma|^{1/2}} \exp\!\left(-\frac{1}{2}(x - \mu)^\top \Sigma^{-1}(x - \mu)\right)

A vector $X$ is multivariate Normal iff every linear combination $a^\top X$ is univariate Normal. This characterization makes the family closed under linear transformations: if $X \sim N(\mu, \Sigma)$ , then $AX + b \sim N(A\mu + b, A\Sigma A^\top)$ .

Conditional distributions are also Normal, with explicit formulas — partition $X$ into $(X_1, X_2)$ and the conditional $X_1 \mid X_2$ is Normal with a closed-form mean and covariance.

Joint Normality matters in finance because portfolio returns and factor models are routinely modeled this way. Real returns are heavier-tailed than Normal, but multivariate Normal is the baseline you build on.

The multivariate Normal $N(\mu, \Sigma)$ is the workhorse of high-dimensional statistics. Its density is

f(x) = \frac{1}{(2\pi)^{k/2} |\Sigma|^{1/2}} \exp\!\left(-\frac{1}{2}(x - \mu)^\top \Sigma^{-1}(x - \mu)\right)

Conditional distributions are also Normal, with explicit formulas — partition $X$ into $(X_1, X_2)$ and the conditional $X_1 \mid X_2$ is Normal with a closed-form mean and covariance.