I am trying to understand the confidence interval for linear regression parameters. At this link Derive Variance of regression coefficient in simple linear regression an answer is provided. However, I did not understand in the derivation fully. Do we consider the regression parameters as random variables?
That is what are the random variables in the following regression: $$ y=\beta_1 x+\beta_0 + \epsilon $$ From what I understand is that just the $\epsilon$ is random variable with mean 0 and variance $\sigma^2$. Considering $\beta_1, x, \beta_0$ as constants $y$ also becomes random variable with mean $\beta_1 x+\beta_0$ and variance $\sigma^2$, from which confidence interval can be calculated.
However,for regression coefficients confidence intervals are also calculated. Can you please clarify the random variables in regression please?