Random unitary beamforming is one of the schemes reducing the amount of feedback information in multiuser diversity techniques with multiple-antenna downlink transmission. In Multiple-Input Multiple-Output (MIMO) systems, throughput performance is greatly improved using AMC (Adaptive Modulation and Coding). Throughput performance is also improved by allocating power among streams appropriately. In random unitary beamforming, transmitter has only partial channel state information (CSI) of each receiver because of reducing the amount of feedback information. Thus, the power allocation scheme needs to work with only partial CSI and low latency. In this paper, we propose a new power allocation scheme for random unitary beamforming assuming a discrete transmission rate with a small amount of feedback information and low latency. Simulation results show that the proposed scheme can improve throughput compared to conventional power allocation scheme.