Multivariate Normal Distribution
Definition
The multivariate normal distribution $N_{p} \left( \mu , \Sigma \right)$ has a probability density function based on the mean vector $\mathbf{\mu} \in \mathbb{R}^{p}$ and the covariance matrix $\Sigma \in \mathbb{R}^{p \times p}$ as follows:
$$ f (\textbf{x}) = \left( (2\pi)^{p} \det \Sigma \right)^{-1/2} \exp \left[ - {{ 1 } \over { 2 }} \left( \textbf{x} - \mathbf{\mu} \right)^{T} \Sigma^{-1} \left( \textbf{x} - \mathbf{\mu} \right) \right] \qquad , \textbf{x} \in \mathbb{R}^{p} $$
- $\mathbf{x}^{T}$ denotes the transpose of $\mathbf{x}$.
Theorems
$$ \begin{align*} \mathbf{X} =& \begin{bmatrix} \mathbf{X}_{1} \\ \mathbf{X}_{2} \end{bmatrix} & : \Omega \to \mathbb{R}^{n} \\ \mu =& \begin{bmatrix} \mu_{1} \\ \mu_{2} \end{bmatrix} & \in \mathbb{R}^{n} \\ \Sigma =& \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} & \in \mathbb{R}^{n \times n} \end{align*} $$ In the statements of the theorems below, unless otherwise specified, $\mathbf{X}$, $\mu$, and $\Sigma$ refer to the same block matrix.
Linear transformation of multivariate normal distribution
The linear transformation $\mathbf{Y} = A \mathbf{X} + \mathbf{b}$ of a random vector $\mathbf{X} \sim N_{n} \left( \mu , \Sigma \right)$ that follows a multivariate normal distribution, given matrix $A \in \mathbb{R}^{m \times n}$ and vector $\mathbf{b} \in \mathbb{R}^{m}$, still follows a multivariate normal distribution $N_{m} \left( A \mu + \mathbf{b} , A \Sigma A^{T} \right)$.
Independence and zero correlation are equivalent in multivariate normal distributions
Let there be a random vector $\mathbf{X} \sim N_{n} \left( \mu , \Sigma \right)$ that follows a multivariate normal distribution. Then, the following holds: $$ \mathbf{X}_{1} \perp \mathbf{X}_{2} \iff \Sigma_{12} = \Sigma_{21} = O $$
Conditional mean and variance of multivariate normal distribution
Given a random vector $\mathbf{X} \sim N_{n} \left( \mu , \Sigma \right)$ that follows a multivariate normal distribution, the conditional probability vector $\mathbf{X}_{1} | \mathbf{X}_{2} : \Omega \to \mathbb{R}^{m}$ still follows a multivariate normal distribution, specifically having the following mean vector and covariance matrix: $$ \mathbf{X}_{1} | \mathbf{X}_{2} \sim N_{m} \left( \mu_{1} + \Sigma_{12} \Sigma_{22}^{-1} \left( \mathbf{X}_{2} - \mu_{2} \right) , \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} \right) $$
Multivariate normality of regression coefficient vectors
The estimators of regression coefficients $\hat{\beta}$ follow a multivariate normal distribution as follows: $$ \hat{\beta} \sim N_{1+p} \left( \beta , \sigma^{2} \left( X^{T} X \right)^{-1} \right) $$
Moment generating function
The moment generating function of $X \sim N_{p} \left( \mu , \Sigma \right)$ is as follows: $$ M_{X} \left( \mathbf{t} \right) = \exp \left( \mathbf{t}^{T} \mu + {{ 1 } \over { 2 }} \mathbf{t}^{T} \Sigma \mathbf{t} \right) \qquad , \mathbf{t} \in \mathbb{R}^{p} $$
Entropy
The entropy of the multivariate normal distribution $N_{p}(\mu, \Sigma)$ is as follows:
$$ H = \dfrac{1}{2}\ln \left[ (2 \pi e)^{p} \left| \Sigma \right| \right] = \dfrac{1}{2}\ln (\det (2\pi e \Sigma)) $$
$\left| \Sigma \right|$ is the determinant of the covariance matrix.
See Also
- Univariate normal distribution: When $p = 1$ followed by $\mu \in \mathbb{R}^{1}$ and $\Sigma \in \mathbb{R}^{1 \times 1}$, the above probability density function exactly becomes the probability density function of the univariate normal distribution.