Brownian terminal-to-maximum ratio
= Brownian terminal-to-maximum ratio
{c}
{title2=$W_1/\max_{0\leq t\leq1}W_t$}
The ratio of the terminal value of a standard <Brownian motion> to its running maximum at time one has density $(2-r)^{-2}$ for $r<1$ and zero otherwise. The <Brownian reflection principle> gives joint density $2(2m-w)\varphi(2m-w)$ for $m>0,w<m$. Substituting $w=rm$ and integrating the Jacobian factor $m$ proves the density; its integral on $(-\infty,1)$ is one.