Marginal distributions of the multinomial normal distribution
Marginal distributions of a multivariate normal distribution are also normal distributions. Let's prove this. See also : Multivariate normal distribution [Wikipedia] The density function of a multivariate normal distribution is given as \[ f(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp\left[-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{\top}\Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})\right] \] where \(\mathbf{x}\in\mathbb{R}^n\) is the random vector, \(\boldsymbol{\mu}\) is the mean vector and \(\Sigma\) is the covariance matrix. By changing the variables \(\mathbf{x} - \boldsymbol{\mu} \mapsto \mathbf{x}\), we can assume the mean is zero without losing generality. So, in the following, we only consider \[ f(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^n|\Sigma|}}\exp\left[-\frac{1}{2}\mathbf{x}^{\top}\Sigma^{-1}\mathbf{x}\right]. \] We need the following theorems from linear algebra. Theorem 1 Let \(A\) be an \(n\times n\) regular, \(D\) be \(m\times m\) regular, \(B\