In this work, the performance analysis of a dual-hop relay transmission system composed of asymmetric radio-frequency (RF)/free-space optical (FSO) links is presented. This work is based on the fact that FSO links are cost-effective, license-free and can provide even higher bandwidths compared to the traditional RF links. More specifically, in this work, we build on the cumulative distribution function presented in  to derive new exact closed-form expressions for the probability density function, moment generating function, and moments of the end-to-end signal-to-noise ratio. We then capitalize on these results to offer new exact closed-form expressions for the higher-order amount of fading, average error rate for binary and M-ary modulation schemes, and ergodic capacity, all in terms of Meijer's G functions. Our new analytical results were also verified via computer-based Monte-Carlo simulation results.