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The power consumption of digital circuits is proportional to the square of operation voltage and the demand for low power circuits reduces the operation voltage towards the threshold of MOSFETs. A weak voltage signal makes circuits vulnerable to noise and the optimization of circuit design requires an accurate noise model. RTN is the dominant noise for modern CMOS technologies. This research focuses on the instability induced by Random Telegraph Noise (RTN) in nano-devices for low power applications, such as the Internet of Things (IoT). RTN is a stochastic noise that can be observed in the drain/gate current of a device when traps capture and emit electrons or holes. The impact of RTN instabilities in devices has been widely investigated. Although progress has been made, the understanding of RTN instabilities remains incomplete and many issues are unresolved. This work focuses on developing a statistical model for characterising, modelling and analysing of the impact of RTN on MOSFET performance, as well as to study the prediction for long-term RTN impact on real circuits. As transistor sizes are downscaled, a single trapped charge has a larger impact and RTN becomes increasingly important. To optimize circuit design, one needs to assess the impact of RTN on circuits, which can only be accomplished if there is an accurate statistical model of RTN. The dynamic Monte Carlo modelling requires the statistical distribution functions of both the amplitude and the capture/emission time (CET) of traps. Early works were focused on the amplitude distribution and the experimental data of CETs has been too limited to establish their statistical distribution reliably. In particular, the time window used has often been small, e.g. 10 sec or less, so that there is little data on slow traps. It is not known whether the CET distribution extracted from such a limited time window can be used to predict the RTN beyond the test time window. The first contribution of this work is three-fold: to provide long-term RTN data and use it to test the CET distributions proposed by early works; to propose a methodology for characterising the CET distribution for a fabrication process efficiently; and, for the first time, to verify the long-term prediction capability of a CET distribution beyond the time window used for its extraction. On the statistical distributions of RTN amplitude, three different distributions were proposed by early works: Lognormal, Exponential, and Gumbel distributions. They give substantially different RTN predictions and agreement has not been reached on which distribution should be used, calling the modelling accuracy into question. The second contribution of this work is to assess the accuracy of these three distributions and to explore other distributions for better accuracy. A novel criterion has been proposed for selecting distributions, which requires a monotonic reduction of modelling errors with increasing number of traps. The three existing distributions do not meet this criterion and thirteen other distributions are explored. It is found that the Generalized Extreme Value (GEV) distribution has the lowest error and meets the new criterion. Moreover, to reduce modelling errors, early works used bimodal Lognormal and Exponential distributions, which have more fitting parameters. Their errors, however, are still higher than those of the monomodal GEV distribution. GEV has a long distribution tail and predicts substantially worse RTN impact. The project highlights the uncertainty in predicting the RTN distribution tail by different statistical models. The last contribution of the project is studying the impact of different gate biases on RTN distributions. At two different gate voltage conditions: one close to threshold voltage |Vth| and the other under operating conditions, it is found that the RTN amplitude follows different distributions. At operating voltage condition, Lognormal distribution has the lowest error for RTN amplitude distribution in comparison with other distributions. The amplitude distribution at close to |Vth| has a longer tail compared with the distribution tail at operating voltage. However, RTN capture/emission time distribution is not impacted by gate bias and follows Log-uniform distribution.