In this theoretical study, the vibration suppression and nonlinear energy transfer, as a function of a dimensionless pendulum length parameter, is investigated for an Inerter Pendulum Vibration Absorber (IPVA) attached to a linear single-degree-of-freedom spring-mass-damper system, subject to white noise excitation. Stochastic differential equations of motion are first developed and integrated to determine the evolution of the response and associated mean and mean square values for long integration times. Dynamic statistical moment equations are then developed, while arc-length continuation is used to track stationary the moments as a function of the pendulum length. Two noise intensity and damping configurations are analyzed and a critical parameter value, in both cases, is found to produce a qualitative change in the system dynamics accompanied by optimal vibration suppression. The results are compared to the response of a linear system without an IPVA to quantify the vibration suppression. Realizations in the time domain are finally calculated to provide validation for the results and gain insight into the changing dynamics of the system as a function of the pendulum length, leading to the discovery of intermittent rotation for sufficiently large pendulum length.