Cell-free (CF) massive multiple-input multiple-output (mMIMO) has been considered as a potential technology for Beyond 5G communication systems. However, the performance of CF mMIMO systems has not been well studied. Most existing analytical work on CF mMIMO systems is based on the expected signal-to-interference-plus-noise ratio (SINR). The statistical characteristics of the SINR, which is critical for emerging applications that focus on extreme events, have not been investigated. To address this issue, in this paper, we attempt to obtain the distribution of SINR in CF mMIMO systems. Considering a downlink CF mMIMO system with pilot contamination, we first give the closed-form expression of the SINR. Based on our analytical work on the two components of the SINR, i.e., desired signal and interference-plus-noise, we then derive the probability density function and cumulative distribution function of the SINR under maximum ratio transmission (MRT) and full-pilot zero-forcing (FZF) precoding, respectively. Subsequently, the closed-form expressions for two more sophisticated performance metrics, i.e., achievable rate and outage probability, can be obtained. Finally, we perform Monte Carlo simulations to validate our analytical work. The results demonstrate the effectiveness of the derived SINR distribution, achievable rate, and outage probability.