Multivariate normal

Summary

• We say that \(X_1,…,X_n\) follow a multivariate normal distribution if the probability density function is given by

\[f\left( x \right)= \frac{1}{{\left( 2π \right)}^{n/2}}{\left| Σ \right|}^{-1/2}exp \left( - \frac{1}{2}{\left( x-μ \right)}'Σ^{-1}\left( x-μ \right) \right)\]

• Here, \(Σ\) is a symmetric, positive definite \(n×n\) matrix of constants, \(\left| Σ \right|\) is the determinant of \(Σ\) , \(μ\) is a \(n×1\) vector of constants and

\[x=\left( \right)\]

• Letting \(X={\left( X_1 X_2 … X_n \right)}'\) we have

\[E\left( X \right)=μ\]

\[Var\left( X \right)=Σ\]

• We write

\[X∼N\left( μ,Σ \right)\]

• If \(X\) follow a multivariate normal distribution then each individual \(X_i\) will follow a normal distribution,

\[X_i~N\left( μ_i,σ_i^2 \right)\]