  We study the influence of inelastic processes on shot noise and the Fano
factor for a one-dimensional double-barrier structure, where resonant tunneling
takes place between two terminals. Most studies to date have found, by means of
various approximate or phenomenological methods, that shot noise is insensitive
to dephasing caused by inelastic scattering. In this paper, we explore the
status of this statement by deriving a general Landaur-B\"uttiker-type formula
that expresses the current noise and Fano factor in a one-dimensional conductor
through inelastic scattering amplitudes. For a double-barrier structure, exact
scattering amplitudes are calculated in the presence of a time-dependent
potential. As an example of dephasing potential, we consider the one induced by
equilibrium phonons. We calculate transmission coefficients of a double-barrier
structure for these two types of phonon-induced dephasing. In the case of
diffusive phase relaxation valid for one dimension phonons, the resonant level
has a Lorentzian shape. For phonons whith high dimensions logarithmic dephasing
realized which leads to an unusual shape of the size-quantized level
characterized by the two energy scales. We further calculate the Fano factor
for these types of dephasing, using exact expressions for inelastic
transmission and reflection amplitudes. It turned out that when an integer
number of levels fall into the energy window of width eV, where V is the
voltage applied to the structure, the Fano factor is really insensitive to
inelastic processes inside the structure and coincides with the prediction of
phenomenological models with an accuracy of small corrections depending on
these processes. On the contrary, at low voltages, when the eV window is
smaller than the level width, this dependence is particularly pronounced and
the phenomenological formula does not work.
