What is the difference between $ y = \alpha+\beta_1x_1+\beta_2x_2 $ and $ y = \alpha+\beta_1x_1+\beta_2x_2+\epsilon $ for a linear regression? I am told that the equations in my question are different but don't really understand why they are different. One said the equation, $ y = \alpha + \beta_1x_1 + \beta_2x_2 $ is deterministic and the other one is probabilistic. Is the statement true? I am not entirely sure if the one who said it is correct or not. Even though the statement is true, I don't really understand what they mean.
 A: Indeed, the equation $y=\alpha + \beta_1 x_1 + \beta_2 x_2$ is deterministic because it means that once you have the values of both predictor variables, you can obtain the value of the response variable with perfect certainty (i.e. deterministically) as expressed by the equation. By way of contrast, the inclusion of the probabilistic noise term $\epsilon$ entails that, aside from the predictor variables, there are random factors that influence the response variable.
A: One shouldn't really write the two equations next to each other, because $y$ has different meaning in them.
The deterministic equation $y = \alpha + \beta_1 x_1 + \beta_2 x_2$ is the expected relation between variables $y$, $x_1$, and $x_2$. In reality such a good relation is never observed: the actual observations deviate from it by the amounts $\epsilon$, which are random, i.e. different from one observation to another:
\begin{equation}
y^{(i)} =  \alpha + \beta_1 x_1^{(i)} + \beta_2 x_2^{(i)} + \epsilon^{(i)}
\end{equation}
where $i = 1...N$ is the index enumerating the observations.
In practice writing many indices complicates reading and one adopts sloppier forms of this expression, as indicated in the question.
