= Polar decomposition of a planar standard normal vector
{title2=$\Theta\perp R^2,\quad\Theta\sim\operatorname{Unif}[0,2\pi),\quad R^2\sim\operatorname{Exp}(1/2)$}
For <independent> <standard normal random variables> $X,Y$, put $X=R\cos\Theta$, $Y=R\sin\Theta$, with $R\geq0$ and $0\leq\Theta<2\pi$. The <polar coordinates> <Jacobian determinant> is $r$, so the <joint probability density> is $r e^{-r^2/2}/(2\pi)$ on $r>0$. It factorizes into a <Rayleigh distribution> density for $R$ and a <continuous uniform distribution> density for $\Theta$. Consequently the radius and angle are <independent>. The <change of variables for probability densities> $S=R^2$ gives $f_S(s)=e^{-s/2}/2$ for $s>0$, the <exponential distribution> of rate $1/2$, also the <chi-squared distribution> with two degrees of freedom. Measurable transformation preserves <independence>, giving $\Theta\perp S$. The angle at $R=0$ is arbitrary on an event of <probability> zero.
Thus a sector with angular length $\ell$ has <probability> $\ell/(2\pi)$, independently of any radial restriction. For instance $Y>|X|$ occupies a sector of angle $\pi/2$ and has <probability> $1/4$.
Back to article page
