Exercise Solution 3.42

Construct the Cholesky matrix z of Σ,


and a matrix u with rows equal to orthonormal eigenvectors of :


We define the change of variables  ~ N3(0,I) for X


and obtain






Multiplying [s4] out:


We complete the squares for terms involving and  to obtain


We have expressed Y as a linear polynomial of three independent random variables:

  • ,
  • ,
  •  ~ N(0,1).