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This correspondence considers a hybrid cognitive radio (H-CR) where the primary transmitter (PT) transmits message to the primary receiver (PR) whenever traffic occurs by allowing interference level I from the secondary transmitter (ST) to the PR, and the ST transmits messages to the secondary receiver (SR) by controlling the transmitting power based on spectrum sensing and interference level I. The ST opportunistically adopts maximum power when it detects that the PT is not transmitting and controls the transmitting power to restrict the interference level from the ST to the PR below I when alerted that the PT is transmitting. The closed-form expressions of the average rates for the H-CR are derived. Based on the analytical results, the detection probability α and interference level I are optimized to maximize the average rate of the ST by guaranteeing the average rate of the PT. Analysis and simulation results demonstrate that the former matches well to the latter and the average rate for the ST of the H-CR is better than that of two existing CRs where the ST transmits messages to the SR with maximum power only when the ST recognizes that the PT is not transmitting or with controlled power to limit an interference level from the ST to the PR below a constant irrespective of the PT's activity.