In this work, we investigate a statistical approach for dynamic spectrum access and radio resource management (RRM) in opportunistic cognitive radio (CR) networks. We propose a distributed MAC protocol for such networks that enables unlicensed users to dynamically utilize the available spectrum while limiting the imposed interference on primary (PR) users. Our proposed protocol is novel in three aspects. First, it does not require CR users to coordinate with PR users. Second, it does not assume any predefined CR-to-PR power mask, and thus can exploit the available spectrum more efficiently. Third, it provides the PR users with a statistical guarantee on the fraction of time that their reception may be corrupted by CR users. To avoid corrupting PR user receptions, the protocol computes the maximum power that a CR transmission can use based on current network conditions. We show how to compute this maximum power by deriving models for the PR-to-CR and PR-to-PR interference. Simulation experiments illustrate that our MAC protocol can satisfy the statistical guarantee for PR users under various user deployment models and traffic loads.