0
$\begingroup$

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 $f_z(z_1,z_2,z_3,z_4)$ ?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ "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 '14 at 18:35
  • $\begingroup$ The wiki assumes $X$ is a row. Here $X$ is a column. Hence the difference. $\endgroup$ – Tom Minka Oct 24 '14 at 15:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.