Difference $Y=Xb+e$, $Y_i=X_ib+e_i$, $y_i=x_ib+e_i$ and $\hat{\ y_i}=x\hat{\ b_i}$ I'm confused about the following four expression used to represent the regression. My intuitions are the following: First: $$Y=Xb+e$$
This is the general linear regression, where random variables $Y$ is explained by a regressors random vector $X$ and $e$ is the difference between $Y$ and its conditional expectation given $X$. Second:  $$Y_i=X_ib+e_i$$
This is the representation of linear regression for a random sample. Where the equation represents an equation system with $n$ equations ($n$ is the size of random sample). $X_i$ and $Y_i$ still are random variables, but represent the random experiment associated to select an individual. Thus, $X_i$ and $Y_i$ aren't realizations yet. Third:  $$y_i=x_ib+e_i$$
Here $x_i$ and $y_i$ aren't random variables, now they are realizations of the random variables $X_i$ and $Y_i$. Fourth:$$\hat{\ y_i}=x\hat{\ b}$$
This is the estimated model. Where $\hat{\ y_i}$ and $\hat{\ b}$ was calculated based on the estimator obtained from the second equation. My question: are my intuitions correct?
 A: Yes. Your intuition is correct.
Let's start with the first and third examples.
Let be $X$ and $Y$ two random variables.
A linear model that relates $X$ to $Y$ is written as
$$
Y=X\beta+\epsilon
$$
This is expressing the linear relationship between the two random variables.
If you have two samples of $X$ and $Y$, $X=\{x_1, x_2, \ldots, x_N\}$, $Y=\{y_1, y_2, \ldots, y_N\}$, the sample linear regression becomes:
$$
y_i = x_i \beta + \epsilon, \quad \forall i \in \{1, 2, \ldots, N\}
$$
This is what one uses to estimate $\beta$, for example.
Example 2. If you have a set of $M$ pairs of random variables $\{(X_1, Y_1), (X_2, Y_2), \ldots, (X_M, Y_M)\}$, then you can express the linear relationship for each of them as:
$$
Y_i=\beta_i X_i + \epsilon_i, \quad i \in \{1, 2, \ldots, M\}
$$
Bear in mind that in general, $\beta$ can be different from one model to another, that's the reason why I wrote $\beta_i$. In the scheme of individuals, as you described, these are called random effects (for hierarchical models).
Example 4. After estimating $\beta$, denoted $\hat{\beta}$, the prediction from the linear model becomes
$$
\hat{Y} = X \hat{\beta}
$$
or
$$
\hat{y}_i = x_i \hat{\beta}, \quad i \in \{1, 2, \ldots, N\}
$$
