Stimulated Brillouin scattering (SBS) induced noise is harmful to long-haul fiber sensing systems. We use the localized fluctuating model to study SBS in a single-mode fiber (SMF) and obtain the relation between the intensity fluctuations and the input power. Experimental results show that relative intensity noise (RIN) is small when the input power is below the SBS threshold and increases dramatically at first and then becomes flat gradually when the input power is above the threshold, which agrees well with the theory. We also use an unbalanced Michelson interferometer with the optical path difference (OPD) of 1 m as well as the phase-generated carrier (PGC) technique to measure the phase noise and find that the curves of phase noise versus input power at different frequencies are similar to that of RIN. We also infer that the increase of the phase noise is mainly due to the broadening of the linewidth caused by SBS.