# Questions Regarding Linear Regression

Are the slope and intercept of a simple linear regression model always normally distributed? Is there ever a difference between the distribution of the estimated slope and intercept and the actual ones? I have only just begun learning about the subject but I am still not clear on the details. A final question: is the least squares method the same as linear regression in that it gives information like the $R^2$? Thanks!

They are if the residuals are independent and normally distributed with common variance. More precise: The linear model is given by $\vec{y} = X\vec{\beta} + \vec{\varepsilon}$, where the design matrix $X$ is $n\times(k+1)$ with $\mathrm{rk}(X)=k+1 <n$ and $\vec{\beta}$ is a vector which contains the intercept and the slopes. Additionally we assume that $\vec\varepsilon \sim N(0, \sigma^2 I)$, that is, we assume that $\vec{y} \sim \mathrm{N}_n(X\vec{\beta}, \sigma^2 I)$. Then we have $\hat{\vec{\beta}}_{ML} \sim \mathrm{N}_{k+1}(\vec{\beta}, \sigma^2(X^T X)^{-1})$, i.e. the maximum likelihood estimator of the intercept and the slopes is multivariate normal.