In this Letter, we experimentally investigate fast temporal intensity dynamics and statistical properties of the cladding-pumped Er/Yb co-doped random Rayleigh feedback fiber laser (EYRFL) for the first time, to the best of our knowledge. By using the optical spectral filtering method, strong and fast intensity fluctuations with the generation of extreme events are revealed at the output of EYRFL. The statistics of the intensity fluctuations strongly depends on the wavelength of the filtered radiation, and the intensity probability density function (PDF) with a heavy tail is observed in the far wings of the spectrum. We also find that the PDF of the intensity in the central part of the spectrum deviates from the exponential distribution and has the dependence on the laser operating regimes, which indicates some correlations among different frequency components exist in the EYRFL radiation and may play an important role in the random lasing spectrum stabilization process.