  Using standard results from statistics, we show that for Gaussian quantum
systems the distribution of a time measurement at a fixed position can be
directly inferred from the distribution of a position measurement at a fixed
time as given by the Born rule. In an application to a quantum particle of mass
$m$ falling in a uniform gravitational field $g$, we use this approach to
obtain an exact explicit expression for the probability density of the
time-of-arrival (TOA). In the long time-of-flight approximation, we predict
that the average positive relative shift with respect to the classical TOA in
case of a zero initial mean velocity is asymptotically given by $\delta =
\frac{q^2}{2} $ when the factor $q\equiv \frac{\hbar}{2m\sigma \sqrt{2gx}} \ll
1$ (semi-classical regime), and by $\delta = \sqrt{\frac{2}{\pi}}q $ when $q\gg
1$ (quantum regime), where $\sigma$ is the width of the initial Gaussian
wavepacket and $x$ is the mean distance to the detector. We also discuss
experimental conditions under which these predictions can be tested.
