I'm working through a proof of the coefficient of linear regression (r) from its verbose form to its concise one. I realize the concept is statistics, but the the proof seems more algebra and possibly calculus-based.
It seems easiest to work backward from the Concise Form to the Verbose one, and to do so I'm using the following definitions:
Doing basic substitution in the Verbose Form I get (didn't substitute for or to keep some semblance of readability):
I'm kind of stuck on what to do with the numerator, which seems to result in distributing a summation to other summations. Since x and y are two "paired" sets n will be the same for all summations and also a constant.
To simplify my question:
Am I able to distribute the like so?:
If so, what would the term resolve to?
Note: I don't want anyone to solve the proof here, I'm just trying to understand how I might be able to resolve the summations. I'd like to work through the proof myself to understand how it works.
Whew, okay first time using LaTeX that took a lot out of me...