 Brownian terminal-to-maximum ratio

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 $2(2m-w)\phi (2m-w)$ for $m>0,w<m$. Substituting $w=rm$ and integrating the Jacobian factor $m$ proves the density; its integral on $(-\infty ,1)$ is one.