In this paper, a slotted ALOHA-based cognitive radio (CR) network is proposed and the throughput performance of the proposed CR network model under Rayleigh fading channels is examined. Our CR network contains two special groups of users, primary users (PUs) and CR users (CRUs), and they are considered to be sharing a time slotted-based common communication channel. While PUs can access the channel at any time owing to their legal right, CRUs can only access the channel when it is not occupied by the PUs. In the network model developed, PUs access the channel utilizing time division multiple access as a medium access control technique, and CRUs can access the channel by exploiting slotted ALOHA as a random access scheme when the channel is idle. In the proposed network model additive white Gaussian noise and Rayleigh channels is considered for comparison reasons. Taking capture effect into account in Rayleigh fading channels, we have obtained new equations for the throughput of the proposed CR network. We have also developed, modeled, and simulated a sample networking scenario by using MATLAB with the aim of validating the analytical throughput results. Simulation results of the proposed network model precisely match with the analytical results obtained under different network load conditions. Furthermore, it is shown that the overall channel exploitation is increased by utilizing the spectrum holes without interfering with the PUs' transmissions.