In spite of the great success of the i-vector/PLDA framework, speaker verification in noisy environments remains a challenge. To compensate for the variability of i-vectors caused by different levels of background noise, this paper proposes a new framework, namely SNR-invariant PLDA, for robust speaker verification. By assuming that i-vectors extracted from utterances falling within a narrow SNR range share similar SNR-specific information, the paper introduces an SNR factor to the conventional PLDA model. Then, the SNR-related variability and the speaker-related variability embedded in the i-vectors are modeled by the SNR factor and the speaker factor, respectively. Accordingly, an i-vector is represented by a linear combination of three components: speaker, SNR, and channel. During verification, the variability due to SNR and channels are marginalized out when computing the marginal likelihood ratio. Experiments based on NIST 2012 SRE show that SNR-invariant PLDA achieves superior performance when compared with the conventional PLDA and SNR-dependent mixture of PLDA.