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Independence gives the joint probability density function $f_{X,Y}(x,y)=4e^{-(x^2+y^2)}/\pi$ on $x,y>0$. The inverse transformation isIts Jacobian matrix has absolute determinant $1/[2(1+v^2)]$. One way to see the determinant is to use polar coordinates: $u=r^2$, $v=\tan \theta$, with $0<\theta <\pi /2$. Then $dxdy=rdrd\theta =dudv/[2(1+v^2)]$. The change-of-variables formula for probability densities givesand zero outside that support. Integrating over the complementary variable givesThese are respectively a unit-rate exponential distribution and a unit-scale half-Cauchy distribution. Their product equals the joint probability density function, so $U$ and $V$ are independent. This radial-ratio independence of two half-normal variables results from the radial symmetry of the positive-quadrant joint probability density function.

Set $\alpha =\arctan c\in (0,\pi /2)$. In polar coordinates, the event is $\sin \Theta >c|\cos \Theta |$, the angular sectorBecause $\Theta$ has the continuous uniform distribution on an interval of length $2\pi$, the probability is the sector's angular length divided by $2\pi$:The symmetry of the standard normal distribution in $Y$ makes the lower sector $Y<-c|X|$ equally likely. Their disjoint union is $|Y/X|>c$, apart from $X=0$, an event of probability zero. Therefore the requested cumulative distribution function isThis is the unit-scale half-Cauchy distribution. At $z=0$ there is no atom, and as $z\to \infty$ the cumulative distribution function tends to one.

The left panel uses $c=1$, giving a sector of angle $\pi /2$ and probability $1/4$. The right panel displays the domain $u<v<2u$ for the next change of variables for probability densities; a vertical section through it determines the conditional range.

If $X,Y$ are independent with density $2e^{-x^2}/\sqrt{\pi }$ on the positive axis, the Jacobian matrix of the inverse transformation has determinant $1/[2(1+v^2)]$. The joint density is $2e^{-u}/[\pi (1+v^2)]$ for $u,v>0$, factoring into an exponential distribution for $U$ and a unit-scale half-Cauchy distribution for $V$. The underlying radial symmetry makes radius and angle independent.