Probability of False Alarm versus SNR
Hi: In statistics we call that the probability of type I error ( rejecting when true ) and type II error ( accepting when false ). The way it's done there is that, once you make an assumption about the distribution of the data ( i.e: normal, t, whatever ), you decide on the null and the alternative, along with what you want the P(type I error ) to be ( say 0.05 ) and then, given the assumption about the distribution, your type II error probability is then known or can be calculated ( sometimes with difficulty ). Your case may be slightly different but I would think that a statistical signal processing book would have a "hypothesis testing" section which would explain what I am describing in more detail.  I hope this helps a little. I put it in the answer section even though it's not really an answer because it was too long for a comment.