Exercise Solution 3.42

Construct the Cholesky matrix z of Σ,

[s1]

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

[s2]

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

[s3]

and obtain

[s4]

where

[s5]

[s6]

[s7]

Multiplying [s4] out:

[s8]

We complete the squares for terms involving and  to obtain

[s9]

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

  • ,
  • ,
  •  ~ N(0,1).

 

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