Expected value and variance of linear functions

Problem

\(ε_1, \ldots ,ε_n\) are \(n\) IID random variables where \(E\left( ε_i \right)=0\) and \(Var\left( ε_i \right)=σ^2\) . \(ε=\left( ε_1, \ldots ,ε_n \right)\) is an \(n×1\) random vector. \(A\) is an \(n×k\) matrix of constants.

  1. Find \(E(ε)\)
  2. Find \(Var\left( ε \right)\)
  3. Find \(E\left( A'ε \right)\)
  4. Find \(Var\left( A'ε \right)\)

Solution

  1. \(E\left( ε \right)=0\) where “ \(0\) ” is an \(n×1\) vector of zeros.
  2. \(Var\left( ε \right)\) is an \(n×n\) matrix. Since \(Var\left( ε_i \right)=σ^2\) for all \(i\) , all the diagonal elements of \(Var\left( ε \right)\) is \(σ^2\) . Since \(ε_1, \ldots ,ε_n\) are independent, \(Cov\left( ε_i,ε_j \right)=0\) if \(i≠j\) . Therefore, every off-diagonal element of \(Var\left( ε \right)\) is zero. Therefore, \(Var\left( ε \right)=σ^2I\) where \(I\) is the \(n×n\) identity matrix.
  3. \(E\left( A'ε \right)=A'E\left( ε \right)=A'0=0\) where the second “ \(0\) ” is a \(k×1\) vector of zeros ( \(A'ε\) is \(k×1\) )
  4. \(Var\left( A'ε \right)=A'Var\left( ε \right)A=A'σ^2I A=σ^2A'A\) . Note that \(A'A\) is \(k×k\) which it should be since \(A'ε\) is \(k×1\) .