Unmanned aerial vehicle (UAV) networks have grown rapidly in recent years and become attractive for various emergence communications scenarios. In this paper, we consider a UAV acting as a relay node to assist wireless transmissions from a base station (BS) to a ground user (GUs). A closed-form expression of outage probability for the BS-GUs transmission via UAV relaying is derived over Rician fading channels. We then formulate an optimization problem to minimize the outage probability of UAV relay communications with a constraint on the total transit power of the BS and GUs. It is proved that our formulated optimization problem is convex and an optimal power allocation solution is found for the outage probability minimization. Simulation results demonstrate that with an increasing power allocation factor, the outage probability initially decreases and then starts to increase, showing the existence of an optimal power allocation solution. Additionally, it is shown that the proposed optimal power allocation scheme significantly outperforms the conventional equal power allocation in terms of the outage probability.