The intrinsic spectral width and intensity dynamics of the acoustic wave generated by the Brillouin random fiber laser were characterized experimentally for the first time. These are important to the understanding of the dynamic noise properties of random fiber lasers based on stimulated Brillouin scattering. We demonstrate that the spectra of the acoustic wave in the gain medium are determined by the spectral convolution of pump light and its Stokes light, which is further an influential factor on the narrow linewidth of the random fiber laser resulting from the distributed feedback of coherent Rayleigh scattering. The power of probe light is weak to ensure the minimum disturbance to the random fiber laser. In the time domain, the intensity of the reflected probe light exhibits a similar output property to that the Brillouin random fiber laser with Gaussian probability distribution when both narrow linewidth lasers (on the order of several kHz) are used as pump light and probe light. In contrast, stochastic noise features with an exponential probability distribution are introduced to the intensity dynamics of the reflected probe light when the linewidth of pump light or probe light is several MHz. The phase noise and intensity noise of the reflected probe light prove that acoustic wave generation and detection is based on a four wave mixing process, which enhances our understanding of the wave coupling in random fiber lasers and gives us a new perspective to understand the fundamental physics of the random lasing process and its noise property.