In this paper, we investigate the so-called noise propagation effect in a mixed radio-frequency/free-space optical (RF/FSO) amplifying-and-forwarding (AF) relaying system that is applied for data transmission in wireless sensor networks. The noise propagation could be essentially severe when battery-charged sensor nodes have very limited transmit power. We provide an exact expression on the cumulative distribution function (CDF) of end-to-end signal-to-noise power ratio (SNR) for a dual-hop mixed RF/FSO AF relaying system. We assume a tightly power-constrained amplifying gain at the relay, which has been usually ignored in existing performance studies for the mixed RF/FSO AF system. It however should be considered to properly evaluate the noise propagation effect especially if the relaying power is not infinite or the sensor has a poor budget in transmit power. We apply the derived exact CDF to evaluate the system performances such as outage probability, average bit-error rate, and ergodic capacity. Numerical investigation is used to justify that the proposed analysis is exactly matched with the simulation and shows that the performance gap caused by the inclusion of the noise propagation effect is significant (about 2-12%) especially when the SNR per hop is in the medium- or the low-SNR ranges (i.e., at 10-20 dB).