I am looking for a way to find the joint pdf of vector $Z=[Z_1,Z_2,Z_3,Z_4]$ where

$Z_1= a_1 X_1^2 + a_2X_1Y_1+ a_3 X_1Y_2 + a_4Y_1^2 + a_5Y_2^2$ $Z_2= b_1 X_1^2 + b_2X_1Y_1+ b_3 X_1Y_2 + b_4Y_1^2 + b_5Y_2^2$ $Z_3= c_1 X_1^2 + c_2X_1Y_1+ c_3 X_1Y_2 + c_4Y_1^2 + c_5Y_2^2$ $Z_4= d_1 X_1^2 + d_2X_1Y_1+ d_3 X_1Y_2 + d_4Y_1^2 + d_5Y_2^2$

where $a_i,b_i,c_i,d_i$ are real numbers and $X_i,Y_i$ are independent zero mean Gaussian random variables. Anybody knows how to find $f_z(z_1,z_2,z_3,z_4)$ ?


1 Answer 1


Matrix algebra is your friend here. Let $$ Z = \begin{bmatrix} Z_1 \\ \vdots \\ Z_4 \end{bmatrix} \qquad X = \begin{bmatrix} X_1 \\ Y_1 \\ Y_2 \end{bmatrix} \\ XX^T = \begin{bmatrix} X_1^2 & X_1 Y_1 & X_1 Y_2 \\ X_1 Y_1 & Y_1^2 & Y_1 Y_2 \\ X_1 Y_2 & Y_1 Y_2 & Y_2^2 \end{bmatrix} $$ The matrix $XX^T$ has a Wishart distribution with 1 degree of freedom and a diagonal scale matrix. $Z$ is a linear transformation of $XX^T$, so its distribution is a linear transformation of a Wishart distribution.

  • $\begingroup$ "The matrix $XX^T$ has a Wishart distribution". What I read from en.wikipedia.org/wiki/Wishart_distribution is $X^TX$ is wishart distribution. Can You please refer to any documents? $\endgroup$
    – upol94
    Oct 23, 2014 at 18:35
  • $\begingroup$ The wiki assumes $X$ is a row. Here $X$ is a column. Hence the difference. $\endgroup$
    – Tom Minka
    Oct 24, 2014 at 15:33

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