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