Non-Orthogonal Multiple Access (NOMA) is emerging as an efficient multiple access technique for the fifth-generation (5G) and beyond 5G (B5G) cellular networks in order to meet the increasing demands on low latency, high reliability, massive connectivity, improved fairness, and high throughput. On the other hand, cooperative relay networks have shown to improve significantly the throughput, the coverage, and the achievable rates of wireless networks, in which the relay nodes assist the communication between the source node and the destination node when the direct channel is low. In this paper, three users’ scenario that simultaneously send information to a base station with the help of a half-duplex relay applying a decode-and-forward (DF) scheme have been studied. Differently from all the existing works and in order to make the work more realistic, it has been assumed that (i) all the channels undergo frequency-selective multipath fading and (ii) the power allocation factors at the transmitting nodes are decided based on the statistical Channel State Information (CSI) not perfect CSI. Under Rayleigh fading channels, expressions of the outage probability are derived and an ad-hoc approach is proposed in order to determine the power allocation factors. Furthermore, the effect of relay location on the outage performance is investigated and the importance of choosing the power allocation factors carefully is demonstrated. Simulation results show that the proper selection of the power allocation factors and the relay location plays a decisive role in uplink NOMA’s performance jointly. Moreover, NOMA always performs better than the Orthogonal Multiple Access (OMA).