Please go through my draft summary below and let me know if my conventions are correct, comprehensible, and non ambiguous.
Simple Linear Regression Model
Let given observed sample set be $\{(x_1,y_1),(x_2,y_2), \cdots (x_N,y_N)\}$.$X,Y$ are random variables that can take on any value $(x_i,y_i)$ within range of sample set.
Population Regression Function, PRF
Given a population $(X,Y)$ we hypothesize underlying population has a regression line as follows. The conditional expectation is
$$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1x & \text{PRF} \end{aligned}$$
The above equation is called Population Regression Function, PRF. Including the error $\varepsilon$, the prediction of dependent variable would be
$$\begin{aligned} & Y = E(Y|x) + \varepsilon & \text{Prediction} \end{aligned}$$
which is called simple linear regression model for population.
RVs: $X, Y|x, \varepsilon$
Parameters $\beta_0, \beta_1, \mu_X, \sigma_X^2, \ \ \mu_{Y|x}, \sigma_{Y|x}^2\ \ \sigma^2$
Sample Regression Function, SRF
Point Estimates from single SRF
Given a sample set $(X,Y)$, we estimate underlying population has a regression line as follows.
\begin{align} & \hat{Y} = \hat{\beta_0} + \hat{\beta_1}x & \text{SRF, Estimator of RV } E(Y|x), \text{ not } Y \\ & \hat{\varepsilon} = Y - \hat{Y} & \text{Estimator of RV } \varepsilon \end{align}
For given sample $(x_i, y_i)$ from sample set $(X,Y)$, a fitted value and residual are $$\begin{aligned} & \hat{y_i} = \hat{Y}(x_i)= b_0 + b_1x_i & \text{Fitted value, Estimate of RV } E(Y|x) \text{ at } x_i \\ & \hat{\varepsilon_i} = y_i - \hat{y_i} & \text{Residual, Estimate of RV } \varepsilon \text{ at } (x_i,y_i) \end{aligned}$$
Using OLS,
$$\begin{aligned} & \hat\beta_1 = \dfrac{\sum_{(x,y)}(y - \overline{Y})(x - \overline{X}) }{\sum_{x}(x - \overline{X})^2} & \text{Slope RV, Estimator of Parameter } \beta_1 \\ & \hat\beta_0 = \overline{Y} - \hat\beta_1\overline{X} & \text{y-intercept RV, Estimator of Parameter } \beta_0 \\ & b_1 = \dfrac{\sum_{i}(y_i - \overline{y})(x_i - \overline{x}) }{\sum_{i}(x_i - \overline{x})^2} & \text{Slope constant, Estimate of Parameter } \beta_1 \\ & b_0 = \overline{y} - b_1\overline{x} & \text{y-intercept constant, Estimate of Parameter } \beta_0 \end{aligned}$$
$\hat{\beta_0},\hat{\beta_1}$ are estimators of $\beta_0,\beta_1$ for any sample set. $b_0,b_1$ are estimates of $\beta_0,\beta_1$ for given sample set
Estimators: $\hat{\beta_0}, \hat{\beta_1}, \hat{X}, \hat{Y}, \hat{\varepsilon}$
Estimates: $b_0, b_1, \overline{x}, s_X^2, \overline{\hat{y}}=\overline{y}, s_{Y|x}^2=s^2$
The entire draft is available here
Ref:
Update 1: As per whuber advice, I also have started separate distinct to the point question, with first one here. Looking forward for further feedback. (For eg, we could either quash this question and continue there, or bring one of them (or first), here modifying this question)
Update 2: Specific split up questions (including first one just for entirety).
1. Doubt about Predictor Variable
2. Is residual an estimate of error
3. Characterizing estimator,estimate separately
4. Why SSE has (n-2)df - not duplicate
