I am studying simple linear regression for the very first time, and I'm having a little trouble understanding something. If someone can clarify this for me and perhaps lead the explanation to a little introduction/motivation behind simple linear models that would be really helpful.
What I've understood is that we have a random variable $Y$ we think is linearly related to a random variable $X$.
But then I've seen the "model" that we use is $Y = \beta_0 + \beta_1 X + \epsilon $. If our assumption was that $X$ and $Y$ are linear, then why did we add an error term? Aren't they exactly linearly related, under our assumption?
I would understand that $y_i = \beta_0 + \beta_1 x_i + \epsilon_i $ was used, perhaps indicating that "While $X$ and $Y$ are perfectly linearly correlated, when we are observing values we have measurement errors and other factors affect this from being a perfect observation.
I thought that maybe our assumption is not that $Y$ and $X$ are exactly linear transforms of each other, but rather just "correlated". If this is a more accurate explanation of our assumption, then the epsilon would make more sense.