= Cauchy exit law from a Brownian half-plane
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{title2=$Y_\tau\overset d=xC$}
Planar <Brownian motion> started at $(-x,0)$ exits the left half-plane with ordinate distributed as $x$ times a standard <Cauchy random variable>. Conditional on the independent horizontal hitting time, the ordinate is normal with variance equal to that time. The <inverse-square Gaussian law of Brownian first passage> turns this into a ratio of independent standard normals. Its exit density is $x/(\pi(x^2+y^2))$.
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