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,

$$X_i~N\left( μ_i,σ_i^2 \right)$$