I'm studying Linear Regression and trying to proof/demonstrate some properties of the parameters. When I started working with the expected value of the slope, I got confused with something. I actually couldn't reach the expected result, which is beta 1.
So, I found a demonstration very close to what I was doing and I was hoping someone could explain to me the part where I got stuck. First, we have beta 1 parameter:
The author then uses Ci to make his demonstration less messy:
The following step is where I got confused. When I see the next part of the demonstration (see picture below), I have no idea why is Ci constant, but Yi isn't! Why is the expected value of Ci equal to Ci, but Yi isn't Yi?
(I even tried this link Expected Value and Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression, but I got nothing. And even in this video: https://www.youtube.com/watch?v=rODUBTRUV0U, the author doesn't explain why is Ci a constant, he just assumes)