Factoring?

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nosit

Junior Member
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Aug 9, 2020
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Hi,
As per my understanding, in the first row the number 2 multiplied by Yi and by (-Xi), only them.
After that, the person factored, and we were left again with Xi and Yi. The - 2 went behind the summation, as it does not vary (it does not have a subscript i).

Is my understanding correct?


1608676503985.png
 
I don't understand why in the second step we have -2.
in the first part we end up with 2Yi, -2(Bo+B1Xi) and -2Xi, so if we factor, before the summation we have to have only 2 and not -2. Am I missing something?
 
Yes. They just pulled a factor of -2 outside the series, and moved the position of the [MATH]X_i[/MATH] in the product that is each term.
 
let [MATH]U_i = (Y_i - (\hat{\beta}_0+\hat{\beta}_1))[/MATH]
we have

[MATH]\sum 2U_i (-X_i) = -2 \sum U_i X_i = -2 \sum X_i U_i = -2 \sum X_i (Y_i - (\hat{\beta}_0+\hat{\beta}_1))[/MATH]
 
Got it. This implies that they multiplied the 2 only by -Xi and then factored. He doesn't need to multiply by all terms...

Right?
 
nosit said:
Hi,
As per my understanding, in the first row the number 2 multiplied by Yi and by (-Xi), only them.
After that, the person factored, and we were left again with Xi and Yi. The - 2 went behind the summation, as it does not vary (it does not have a subscript i).

Is my understanding correct?


View attachment 23935
Click to expand...
No, The 2 is being multiplied by yi-(B0 + BiXi) OR (-Xi)

Consider 2*3*4. You do NOT multiply the 2 by the 3 AND 4. You can multiply the 2 by 3 and multiply that result by 4 OR you can multiply the 2 by 4 and then multiply that result by 3.

Basically you have three factors that are being multiplied. Two of them multiply out to -2X. Since the -2 has nothing to do with what we are summing over, which is i, we can factor that out in front.
 
nosit said:
Got it. This implies that they multiplied the 2 only by -Xi and then factored. He doesn't need to multiply by all terms...

Right?
Click to expand...
You need to multiply all the factors (not terms) but you do not multiply the 2 by each of the other two factors!
 
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