how can I show that given $ \textbf{y} =(y_1,y_2)^T \sim N (\textbf {0} ,\Sigma ) $
$$(\textbf{y}^t\Sigma^{-1}\textbf{y} - \frac{y_1^2}{\sigma_{11}}) \sim \chi^2_1$$ where $ \Sigma =(\sigma_{i,j})$
I have tried considering moment generating functions. Is it true that $\textbf{y}^t\Sigma^{-1}\textbf{y}$ is chi squared with 2 degrees of freedom and that $ \frac{y_1^2}{\sigma_{11}}$ is chi squared with 1 df?
Thanks for your help