According to DRS4's datasheet, the random noise is 0.35mVrms. Is this the input equivalent noise voltage? It is computed over the 0-950MHz frequency band?
You cannot compare the DRS4 noise directly with an amplifier for example. The noise mainly comes from variations of the charge injection into the storage cells, and some noise during the readout process, which happens in a completely different frequency domain than the sampling.
So what I did is to keep the inputs open, measure a 1024-bin waveform, and compute the RMS of this waveform. So I believe that this is kind of equivalent noise voltage from 1-950 MHz. It does not start from zero since very low frequency noise (like 50 Hz) just causes a baseline shift and does not influence the RMS, but this is not so important since in most applications people do an event-by-event baseline subtraction to get rid of low frequency noise in their apparatus. The 0.35 mV RMS also depend on the electronics around the chip. On our USB evaluation board the noise it typically smaller (0.31 mV RMS), while in some VME board we measure 0.42 mV RMS. If you do the perfect analog design around the chip, you can maybe push this maybe even lower.