This paper considers random multiple access in a network where only a small portion of users have data to forward and transmit packets in each time slot because the user activity ratio is not high in practice. For this reason, the access point (AP) has to not only identify the users who transmitted but also decode the received data codewords. Exploiting the sparsity of transmitting users, Lasso, which is a well-known practical compressed sensing algorithm, is applied for efficient user identification. The compressed sensing algorithm enables the AP to handle more users than the conventional random multiple access schemes do. We develop distributed scheduling methods for maximizing the system sum throughput, and we analyze the corresponding optimal throughput for three different cases of channel knowledge, i.e., the channel state information at the transmitter (CSIT), the channel state information at the receiver (CSIR), and the imperfect channel state information at the receiver (ImCSIR). We also derive the closed-form expressions of asymptotically optimal scheduling parameters and the corresponding maximum sum throughput for each CSI assumption. The results show the effects of system parameters on the sum throughput and provide useful insights on using compressed sensing for throughput maximization in random multiple access schemes.