  Exact analytical solutions of the time-dependent Schr\"odinger equation with
the initial condition of an incident cutoff wave are used to investigate the
traversal time for tunneling. The probability density starts from a vanishing
value along the tunneling and transmitted regions of the potential. At the
barrier width it exhibits, at early times, a distribution of traversal times
that typically has a peak $\tau_p$ and a width $\Delta \tau$. Numerical results
for other tunneling times, as the phase-delay time, fall within $\Delta \tau$.
The B\"uttiker traversal time is the closest to $\tau_p$. Our results resemble
calculations based on Feynman paths if its noisy behaviour is ignored.
