  We calculate analytically the probability density $P(t_m)$ of the time $t_m$
at which a continuous-time Brownian motion (with and without drift) attains its
maximum before passing through the origin for the first time. We also compute
the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the
driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim
t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In
presence of a drift towards the origin, $P(t_m)$ decays exponentially for large
$t_m$. The results from numerical simulations are in excellent agreement with
our analytical predictions.
