Suppose X and Y are independent, zero-mean Gaussian random variables with variances of $$σ_X^2 ,σ_Y^2$$ Let $$Z = X^2 + Y^2 , W = Z^2-Y^2$$
How do you find the joint pdf of Z and W?
I found the pdf of Z by scaling the Chi-squared distribution and got $$fz(z) = \frac{1}{2σ_X^2σ_Y^2} exp(-\frac{z}{(σ_X^2σ_Y^2)^2})$$
But I can't find W, or the joint pdf