Inverse-square Gaussian law of Brownian first passage
= Inverse-square Gaussian law of Brownian first passage
{title2=$\tau_x\overset d=x^2/N^2$}
The <Brownian reflection principle> gives $\mathbb P(\tau_x\leq t)=\mathbb P(|N|\geq x/\sqrt t)$ for a standard normal <random variable> $N$. This is the <cumulative distribution function>[distribution function] of $x^2/N^2$. An <independent> <Brownian motion> observed at this <hitting time> has the <Cauchy distribution> with scale $x$.