Linear function of a normal random vector

Problem

\(X=\left( X_1, \ldots ,X_k \right)\) is a \(k×1\) random vector where each \(X_i\) follows a standard normal distribution and where all random variables in \(X\) are independent. We say that \(X\) follows a multivariate standard normal and we write

\[X \sim N(0,I)\]

where “0” is \(k×1\) and \(I\) is the \(k×k\) identity matrix.

Let \(Σ\) be an arbitrary symmetric positive definite \(k×k\) matrix. We say that the \(k×k\) matrix \(Σ^{1/2}\) is the “square root” of \(Σ\) if \(Σ^{1/2}Σ^{1/2}=Σ\) . At this point, we need not get into how \(Σ^{1/2}\) is calculated. Suffice to say that it will exist and it will be symmetric if \(Σ\) is a symmetric positive definite matrix. Let \(μ\) be an arbitrary \(k×1\) vector.

Consider the \(k×1\) random vector

\[Σ^{1/2}X+μ\]

1. Show that \(E\left( Σ^{1/2}X+μ \right)=μ\)
2. Show that \(Var\left( Σ^{1/2}X+μ \right)=Σ\)

Learning point:

• For a random variable \(X\) following standard normal distribution, \(X \sim N\left( 0,1 \right)\) , we have the important result that for arbitrary constants \(μ\) and \(σ>0\) ,

\[σX+μ \sim N\left( μ,σ^2 \right)\]

• It can be shown that any linear combination of random variables following a normal distribution will also be normal. Therefore, all the random variables in \(Σ^{1/2}X+μ\) will be normal. We say that \(Σ^{1/2}X+μ\) follows (or is) a multivariate normal . Since \(E\left( Σ^{1/2}X+μ \right)=μ\) and \(Var\left( Σ^{1/2}X+μ \right)=Σ\) we write

\[Σ^{1/2}X+μ \sim N\left( μ, Σ \right)\]

• which is the multivariate correspondence to the result in the single variable result in the first bullet.