Hello,

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.

Verbose Form:

Concise Form:

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...