Notice that the $\hat{f}$ don't have an error term, $\epsilon$, as would be expected for regression models. Why don't these models have an error term? I am currently reading some notes on linear and polynomial regression. The notes say the following:

The linear model is
$$\hat{f}_L (X) = \beta_0 + \beta_1 X$$
The quadratic model is
$$\hat{f}_Q(X) = \beta_0 + \beta_1 X + \beta_2 X^2$$

Notice that the $\hat{f}$ don't have an error term, $\epsilon$, as would be expected for regression models. Why don't these models have an error term?

For a linear model there are two equivalent ways to write the model:
$$Y = X\beta + \epsilon$$
and
$$E(Y \mid X) = X\beta$$
Where $\epsilon$ is the error term and $E()$ denotes expected value (long-term average).  $X\beta$ is the linear predictor, i.e., weighted sum of predictors.  When writing the model in terms of expected values, the error term need not be included.
For Gaussian errors (the usual linear model) I prefer an all-encompassing statement of the model.   Here $\Phi$ is the Gaussian cumulative distribution function and $\sigma$ is the residual standard deviation.
$$\Pr(Y \leq y \mid X) = \Phi \left( \frac{y - X\beta}{\sigma} \right)$$
If the model is not a linear model, there need not be an error term anywhere.  For example when $Y$ is binary the model is almost always stated as a probability statement similar to the above probability statement.
