Now coming back to the point, it is clear that cualitatively the CMBR matches with the "Big Bang" theory which says that the Universe was getting colder and colder and with time some "low" background radiation is expected. That's fine, my point was more quantitative in that how it could be demonstrated that the "energy level" of the radiation (as you said and I mean temperature) would be predicted precisely by the theory. As I said I think this haven't been done.
You are incorrect in your thinking.