This research investigated a non-orthogonal multiple access (NOMA) system with relay in downlink cellular communication. We assumed that the communication between the base station (BS) and the user is via a half-duplex relay based on a decode-and-forward (DF) protocol. There is also a direct link in this cooperative NOMA system. The exact closed-form expression of user outage probability based on the cooperative NOMA network was derived through the analysis of this model. The outage probability when multiplexing two users were approximated in this NOMA system to analyze the system performance more deeply. The results showed that, compared to the orthogonal multiple access (OMA) system, the cooperative NOMA system performed better outage performance and greater diversity gain. Through simulations, we verified the influence of the relay position on outage performance. Meanwhile, the closer to the source node, the lower probability of outage. These results provide a theoretical basis for practical applications in communications.