Multivariate Normal Distribution

The multivariate normal distribution generalizes the bell curve to multiple dimensions. It is defined by a mean vector \( \boldsymbol{\mu} \) and a covariance matrix \( \boldsymbol{\Sigma} \), and its shape is symmetric and elliptical in \( \mathbb{R}^n \).

If a random vector \( \mathbf{X} = (X_1, X_2, \ldots, X_n) \) follows a multivariate normal distribution, we write

\[\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma}).\]

One key property is that any linear combination of the components of a multivariate normal vector is itself normally distributed. That is, for any constant vector \( \mathbf{a} \), the scalar \( \mathbf{a}^\top \mathbf{X} \) is normally distributed.

This property makes the multivariate normal distribution particularly useful in finance and portfolio theory, where asset returns are often modeled jointly. It also arises naturally when multiple variables are influenced by shared sources of randomness, due to the Central Limit Theorem.