  The effect of electron-phonon coupling on the current noise in a molecular
junction is investigated within a simple model. The model comprises a 1-level
bridge representing a molecular level that connects between two free electron
reservoirs and is coupled to a vibrational degree of freedom representing a
molecular vibrational mode. The latter in turn is coupled to a phonon bath that
represents the thermal environment. We focus on the zero frequency noise
spectrum and study the changes in its behavior under weak and strong
electron-phonon interactions. In the weak coupling regime we find that the
noise amplitude can increase or decrease as a result of opening of an inelastic
channel, depending on distance from resonance and on junction asymmetry. In
particular the relative Fano factor decreases with increasing off resonance
distance and junction asymmetry. For resonant inelastic tunneling with strong
electron-phonon coupling the differential noise spectrum can show phonon
sidebands in addition to a central feature. Such sidebands can be observed when
displaying the noise against the source-drain voltage, but not in noise vs.
gate voltage plots obtained at low source-drain bias. A striking crossover of
the central feature from double to single peak is found for increasing
asymmetry in the molecule-leads coupling or increasing electron-phonon
interaction. These variations provide a potential diagnostic tool. A possible
use of noise data from scanning tunneling microscopy experiments for estimating
the magnitude of the electron-phonon interaction on the bridge is proposed.
