Regression is the study of how the distribution of response varies as the values of its predictors change. Regression models are employed in almost every area of science and engineering and are essential tools for analyzing data. Sometimes obtained data may be in the form of fuzzy uncertainty along with the associated probabilities. As such, it may be interesting and required to develop the corresponding regression models to figure out the process. Accordingly, in this work, the perception of fuzzy random variables (FRV) and associated expectations are discussed in the regression models. Linear and quadratic fuzzy random regression models have been investigated based on fuzzy random data to approximate the fuzzy coefficients. Trapezoidal fuzzy numbers have been considered to express both fuzzy input and fuzzy output variables. Finally, some problems have also been solved to illustrate the application of the current approach.