I'm dealing with some cdf problems. Given two statistically independent standard normal distributions $X$ and $Y$, let $W=X^2+Y^2$, and $Z=X-Y$. I wish to find the cdf $F_{W,Z}(w,z)$.
My strategy starts with the Jacobian. $J(x,y)=-2(x+y)$ in this case. Also, by some algebra, since $x+y=\sqrt{|w^2+w-z^2|}$ Hence the joint pdf of $W$ and $Z$ is $$f_{W,Z}(w,z)=\frac{1}{2}\big(\frac{1}{2\pi}\big)e^{-\frac{w^2}{2}}\frac{1}{\sqrt{|w^2+w-z^2|}},$$ given that $$f_{X,Y}(x,y)=\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$$ However, The integral (if integrated with z first) $$F_{W,Z}(w,z)=\int_{-\infty}^z\int_{-\infty}^w \frac{1}{2}\big(\frac{1}{2\pi}\big)e^{-\frac{a^2}{2}}\frac{1}{\sqrt{|a^2+a-b^2|}} dbda$$ will contain an $\arcsin$, which makes $F_{W,Z}(w,z)$ doesn't looks like it has a closed form.