Suppose X, Y are independent normal $N(0,\sigma^2)$.
Suppose we perform bivariate transformation such that
$U=X^2+Y^2$, $V=X/\sqrt{U}$.
My question is how to get the support of $U,V$.
$U$ : it is obvious that it is between 0 and infinity.
V : I am not sure how it is between -1 and 1.
Can someone explain in detail? Thanks.
Visualize $X$ and $Y$ as corresponding to a random ordered pair $(X,Y) \in \mathbb R^2$ in the Cartesian coordinate plane. Then $\sqrt{X^2 + Y^2}$ is the distance of $(X,Y)$ to the origin $(0,0)$. Then $X/\sqrt{X^2 + Y^2}$ is the ratio of the $X$-coordinate to this distance; i.e., it is the cosine of the angle that the vector $(X,Y)$ forms with the positive $X$-axis.