PDF of sum of independent Gaussian variables

Question

I am looking for deriving the pdf of $Z$ where

$Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$,

where $X_i$ and $Y_i$ are independent, zero mean Gaussian random variables and $a_i$ and $b_i$ are real numbers. Does this follow any standard distribution ? How to find PDF of $Z$ ?

Answer 1

This is generalized $\chi^2$ distribution. You can define as a stacked vector: $v=(X_i;Y_j)$, then represent $Z=vAv$, the rest is like in the Wikipedia link