Is n*Sum(XiYi) - Sum(XiYi) = Sum(Xi - Xbar)(Yi - Ybar)?

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retspool
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Linear models is the topic

And i believe my professor wrote this n*Sum(XiYi) - Sum(XiYi) = Sum(Xi - Xbar)(Yi - Ybar)

i was hoping if some one could tell me if this is true of if it is my bad in typing down.

If it is true then can some one please tell me how?

I've attached an image i'd appreciate it if someone could explain to me the last 3 steps.

Thanks
 
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And this too. I've written simplifies at the center of the paper, i don't know how they get the Betas.


Thanks
 
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Let <x> and <y> denote the x and y averages (you called them Xbar Nd Ybar). You want S = sum{(xj -<x>)(yj - < y>): j=1...n}. The nth term, expanded out, is xj*yj - <x>*yj - <y>*xj + <x>*<y>. Now sum <x>*yj = <x>*sum yj = n*<x>*<y>, etc., so we end up with S = sum(xy) - 2*n*<x>*<y> + n*<x>*<y> = sum(xy) - n<x><y>.

RGV