Multivariate Normal Distribution - Marginal Distributions

Marginal Distributions

To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.

Example

Let x = be multivariate normal random variables with mean vector μ = and covariance matrix Σ (standard parametrization for multivariate normal distributions). Then the joint distribution of x′ = is multivariate normal with mean vector μ′ = and covariance matrix $\boldsymbol\Sigma' = \begin{bmatrix} \boldsymbol\Sigma_{11} & \boldsymbol\Sigma_{13} \\ \boldsymbol\Sigma_{31} & \boldsymbol\Sigma_{33} \end{bmatrix}$ .

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