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Efficient design of spectrum-sensing and power control plays a key role in maximizing the throughput of a cognitive radio (CR) network while protecting the PUs from undue interference. In this paper, we consider the problem of joint spectrum-sensing-duration design and power control in a point-to-point CR link. The objective of the proposed solution is to maximize the CR achievable throughput while keeping the CR interference to the primary user (PU) at or below a preset interference threshold. The joint spectrum-sensing design and power control is a non-convex optimization problem, hence, obtaining the optimal solution may not be always feasible. In this paper, we formulate the joint sensing-duration design and power control problem as a two-stage stochastic program with recourse. Afterward, we derive the formulas for the achievable throughput as a function of the sensing duration. Moreover, we investigate the effect of different levels of knowledge of the interference channel state information (CSI) on the achievable throughput and the optimal sensing duration.