In this paper, we investigate the problem of power allocation in cognitive underlay networks, where a secondary user (SU) is allowed to coexist with a primary user (PU). We consider three transmission models for the secondary link: (i) one-way transmission with relay assisted, (ii) two-way transmission with a direct link, and (iii) two-way transmission with relay assisted. In conventional interference-limited cognitive networks, the instantaneous channel state information (CSI) of a PU is required to suppress SU's transmit power to guarantee the quality of service (QoS) of the PU, which increases the feedback burden in practice. To tackle this issue, in this article we take primary outage probability as a new criterion to measure the QoS of the PU, where only the statistical CSI of the PU is required. Firstly, we derive the primary outage constraints for the three models, respectively. Then, with the newly obtained constraints, we formulate optimization problems to maximize the channel rate of the SU. Finally, we derive the optimal solutions for power allocation with respect to different parameters, respectively. Simulation results verify the performance improvement of the proposed schemes.