In linear regression or Gaussian-Markov framework,
$$Y = \beta_0 + \beta_1X+\epsilon, $$
we usually assume $X$ is non-random. In our statistics class, professors sometimes simply mention like "here $X$ is nonrandom". I wonder what happens if $X$ is assumed to be a random variable?
Let's assume $\epsilon$ is from a normal distribution and is independent of $X$, and $X$ has mean zero.