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In this paper, we investigate a novel slotted ALOHA-based distributed access cognitive network in which a secondary user (SU) selects a random subset of channels for sensing, detects an idle (unused by licensed users) subset therein, and transmits in any one of those detected idle channels. First, we derive a range for the number of channels to be sensed per SU access. Then, the analytical average system throughput is attained for cases where the number of idle channels is a random variable. Based on that, a relationship between the average system throughput and the number of sensing channels is attained. Subsequently, a joint optimization problem is formulated in order to maximize average system throughput. The analytical results are validated by substantial simulations.