The variance of a random vector

Summary

Definition

• \(X\) is an \(n×1\) random vector with expected value \(μ\) .
• Let \(Z=\left( X-μ \right){\left( X-μ \right)}'\) which is an \(n×n\) matrix. We define the variance of a random vector \(X\) as the \(n×n\) matrix

\[\text{Var}\left( X \right)=E\left( Z \right)=E\left[ \left( X-μ \right){\left( X-μ \right)}' \right]\]

• We have

\[Var\left( X \right)=\begin{pmatrix} σ_1^2 & σ_{1,2} & \cdots & σ_{1,n} \\ σ_{2,1} &σ_2^2 & \cdots & σ_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ σ_{n,1} & σ_{n,2} & \cdots & σ_n^2 \end{pmatrix}\]

• where \(σ_i^2=Var\left( X_i \right)\) and \(σ_{ij}=Cov(X_i,X_j)\) .

Results

• If \(c\) is an \(n×1\) vector of constants then

\[Var\left( c'X \right)=c'Var\left( X \right)c\]

• which is a number.
• If \(A\) is an \(m×n\) matrix of constants then

\[Var\left( AX \right)=AVar\left( X \right)A'\]

• which is an \(m×m\) matrix of constants