Expected value and variance for random vectors

Problem

\(X\) is an \(n×1\) random vector, \(X=\left( X_1, \ldots ,X_n \right)\) . The expected value of \(X\) is denoted by \(μ\) . \(A\) is an \(m×n\) matrix of constants.

a) If \(Y=AX\) then \(Y\) is a _______ random vector.

b) Prove that

\[E\left( Y \right)=Aμ\]

• Note that this holds even if \(m=1\) and \(A\) is \(1×n\) . In this case, we typically denote \(A\) by \(a'\) instead such that \(a\) is \(n×1\) . We then have

\[E\left( a'X \right)=a'E\left( X \right)\]

• Similar results hold when \(X\) is an \(n×m\) matrix of random variables. In this case, \(E\left( X \right)\) is an \(n×m\) matrix containing all the expected values. If \(B\) is an \(k×n\) matrix of constants then

\[E\left( BX \right)=BE\left( X \right)\]

• and if \(C\) is an \(m×l\) matrix of constants then

\[E\left( XC \right)=E\left( X \right)C\]

• The reason these results hold is that matrix multiplication is inherently a linear operator.

c) Prove that

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

This holds even if \(m=1\) and \(A\) is \(a'\) . We then have

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

Both sides of this expression are scalars \(1×1\) .