Multivariate random variable

Multivariate normal distribution

A random vector ⃗𝐗 :𝜉 →ℝ𝑘 has a multivariate normal distribution iff every linear combination of 𝑋𝑗 has a normal distribution, prob i.e. ⃗𝐯 ⋅⃗𝐗 ∼N(𝜇,𝜎2) for some 𝜇,𝜎2 for any ⃗𝐯 ∈ℝ𝑘. 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=⎡⎢ ⎢⎣Cov⁡[𝑋1,𝑋1]⋯Cov⁡[𝑋1,𝑋𝑘]⋮⋱⋮Cov⁡[𝑋𝑘,𝑋1]⋯Cov⁡[𝑋𝑘,𝑋𝑘]⎤⎥ ⎥⎦

the Joint probability density function is given by

𝑓⃗𝐗(⃗𝐱)=det(2𝜋Σ)−1/2exp⁡(−12(⃗𝐱−⃗𝜇)𝖳Σ−1(⃗𝐱−⃗𝜇))

Bivariate case

In the bivariate case, we have

𝑓𝑋1,𝑋2(𝑥1,𝑥2)=1√(2𝜋)2𝜎21𝜎22(1−𝜌2)exp⁡(−12(1−𝜌2)((𝑥1−𝜇1𝜎1)2−2𝜌(𝑥1−𝜇1𝜎1)(𝑥2−𝜇2𝜎2)+(𝑥2−𝜇2𝜎2)2))

where 𝜌 =𝜎1,2/√𝜎21𝜎22 =Corr⁡[𝑋1,𝑋2].

Properties

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

---

tidy | en | SemBr