We analyze a cognitive radio network where the primary users (PUs) and cognitive radio (CR) devices are distributed over the two-dimensional plane according to two independent homogeneous Poisson processes. Any CR that lies within the detection region of some PU switches to a different channel in order to prevent causing harmful interference at the PU. Using the concepts of stochastic geometry, we study the characteristics of the interference caused by the PUs and the CRs to a given PU. Further, these results are used to obtain tight upper and lower bounds for the success probability at the PU; defined as the probability that the signal-to-interferenceplus- noise ratio (SINR) is beyond a certain operating threshold.