Jeff,
Have a look at the radon progeny chart:
http://www.ccnr.org/radon_chart.html
The radioactive dust is because Radon seeps into the atmosphere from Uranium decay in the ground/walls and then decays into its progeny (all of which are solid). These solids are however all formed as single atoms and they get hoovered up by dust particles which they stick to.
If most of the dust is old (say > 1 year) you can see that the chain to Pb-210 will have decayed and what you get left is (from each Pb-210 decay with 22 year half-life) 2 betas and one alpha with a very long half-life. The alpha may not easily be detectable.
If the dust is newer (say less than a few hours) you get dust particles with newly hoovered up elements from the air. That would include Pb-214, with 2 betas and an alpha decay with a 30 min or so combined half-life.
The "newer" decay chain will be 20 years/30 minutes = 100,000 X more active than the "older" decay chain so I guess that is what you see because the "old" dust is not picking up radioactive material much when it is sitting on the floor, so you do not get 100,000 X more material even if the dust has been sitting there for 20 years (and of course if you ever dust the basement that will not be the case!). Dust in the air will pick up new radon products, but once the dust is no longer moving through the air this pickup process will stop and the radioactivity in it will decay with approx a 30 min half-life. Although it decays to Pb-210, and this having a half-life of 22 years stays around, the activity from the Pb-210 is 100,000 X less than from the Pb-214 and so can I think be ignored - there is not so very much more Pb-210.
So this demystifies the issue. What you will see mainly is the decay products from Pb-214->Bi-215->Po-214. There should be:
2 X beta
1 X 8 Mev alpha
Best wishes, Tom
PS - this is my best understanding but I'm no expert here, so could be wrong.
PPS Axil - your posts above are off topic for this discussion, which is about Jeff measuring his background radioactivity from dust. It is an important topic, not least for the health of Jeff and possibly his family.