What is the statistical relationship between radioactivity and mutation rate?
According to this wiki page , the average background radiation is 3.01 milli-Sievert per year (including natural and artificial sources). This equals 0.301 rad.

I found a short letter to nature that says the average forward mutation rate in human is 2.6 * 10^-7 per locus per rad = 2.6 mutations per ten million bases. Also it say that this mutation rate is quite uniform among species. The size of the human genome is approximately 3.2 gigabases.
So doing a quick maths: (2.6*3.2*100)*0.301 = 250.432 mutations per year per human. This is an approximation because I rounded the size of the human genome to 3.2 gigabases. Also this doesn't mean that we actually acquire this amount of mutations because we have repair mechanisms.