In real-world regression analysis, statistical data may be linguistically imprecise or vague. Given the co-existence of stochastic and fuzzy uncertainty, real data cannot be characterized by using only the formalism of random variables. To address regression problems in presence of such hybrid uncertain data, fuzzy random variables are introduced in this study, and serve as an integral component of regression models. A new class of fuzzy regression models based on fuzzy random data is built, and is called the fuzzy random regression model (FRRM). First, a general fuzzy regression model for fuzzy random data is introduced. Then, using expectations and variances of fuzzy random variables, σ-confidence intervals are constructed for fuzzy random input-output data. The FRRM is established based on the σ-confidence intervals. The proposed regression model gives rise to a non-linear programming problem which consists of fuzzy numbers or interval numbers. Since sign-changes in the fuzzy coefficients modify the entire programming structure of the solution process, the inherent dynamic non-linearity of this optimization makes it hard to exploit the techniques of linear programming or classical non-linear programming. Therefore, we resort to some heuristics. Finally, an illustrative example is provided.