By treating the harmonic basis quanta of a vibrationally excited molecule as a Bose gas, we derive a statistical vibrational triangle rule local random matrix model (BSTR) for the molecular Hamiltonian, which includes the minimal features required for a description of intramolecular vibrational energy redistribution (IVR). This model incorporates spectroscopic parameters such as vibrational frequencies, as well as low and high order anharmonic couplings. It is detailed enough to roughly represent specific molecules, and sufficiently general to yield generic IVR behavior on the energy shell. The matrix fluctuation-dissipation theorem is used as a computational tool to study the IVR spectra of large molecular systems represented by the BSTR. All classes of behavior from isolated resonances to quantum ergodicity are observed. The dependence of various molecular properties (IVR rate, dilution factor sigma, fraction of occupied phase space F, line spacings, and other statistics) on the available energy, anharmonic coupling strength and state density is studied and compared to experimental results.