The simple linear regression model posits that the response $Y_i$ is related to the predictor $x_i$ by $Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i = 1, \ldots, n$ where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon_1, \ldots, \epsilon_n$ are independent errors with $E[\epsilon_i] = 0$ and $\operatorname{Var}(\epsilon_i) = \sigma^2$. The parameters $\beta_0$, $\beta_1$, and $\sigma^2$ are unknown.

The ordinary least squares (OLS) estimators minimize the sum of squared residuals: $(\hat{\beta}_0, \hat{\beta}_1) = \arg\min_{b_0, b_1} \sum_{i=1}^n (y_i - b_0 - b_1 x_i)^2$ The solution is: $\hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} = \frac{S_{xy}}{S_{xx}}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$