Multivariate normal distribution

A random vector $⃗ 𝐗 : 𝜉 \to ℝ 𝑘$ has a multivariate normal distribution iff every linear combination of $𝑋 𝑗$ has a normal distribution, #m/def/prob i.e. $⃗ 𝐯 \cdot ⃗ 𝐗 \sim N (𝜇, 𝜎 2)$ for some $𝜇, 𝜎 2$ for any $⃗ 𝐯 \in ℝ 𝑘$ . Such a distribution is fully specified by the means and variances of each component, and the covariance of every pair of components. Packaging this information into a mean vector $⃗ 𝜇$ and a covariance matrix

Σ = ⎡ ⎢ ⎢ ⎣ C o v [𝑋 1, 𝑋 1] \dots C o v [𝑋 1, 𝑋 𝑘] ⋮ ⋱ ⋮ C o v [𝑋 𝑘, 𝑋 1] \dots C o v [𝑋 𝑘, 𝑋 𝑘] ⎤ ⎥ ⎥ ⎦

the Joint probability density function is given by

𝑓 ⃗ 𝐗 (⃗ 𝐱) = d e t (2 𝜋 Σ) - 1 / 2 e x p (- 12 (⃗ 𝐱 - ⃗ 𝜇) 𝖳 Σ - 1 (⃗ 𝐱 - ⃗ 𝜇))

Bivariate case

In the bivariate case, we have

𝑓 𝑋 1, 𝑋 2 (𝑥 1, 𝑥 2) = 1 \sqrt (2 𝜋) 2 𝜎 21 𝜎 22 (1 - 𝜌 2) e x p (- 1 2 (1 - 𝜌 2) ((𝑥 1 - 𝜇 1 𝜎 1) 2 - 2 𝜌 (𝑥 1 - 𝜇 1 𝜎 1) (𝑥 2 - 𝜇 2 𝜎 2) + (𝑥 2 - 𝜇 2 𝜎 2) 2))

where $𝜌 = 𝜎 1, 2 / \sqrt 𝜎 21 𝜎 22 = C o r r [𝑋 1, 𝑋 2]$ .

Properties

Any subvector of a multivariate normal vector is multivariate normal.
The concatenation of two independently distributed multivariate normals is multivariate normal.

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