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Least Squares Derivation—Simple Algebraic Simplification

  1. Mar 15, 2015 #1

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    Hey, PF

    I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8.

    [Harvard.edu]

    More specifically, why is the denominator a difference of two terms? Aren't the terms in the denominator summed in the prior step?

    I would expect the answer to be

    $$
    b_1=\dfrac{\displaystyle \sum_{\textrm{i=1}}^{n}y_ix_{i}-\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}y_i\sum_{\textrm{i=1}}^{n}x_{i}\right)}{\displaystyle\sum_{\textrm{i=1}}^{n}x_{i}^2+\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}x_{i}\right)^{2}}
    $$

    Note: I'm no statistician, but I thought you guys might be more familiar with this derivation.
     
  2. jcsd
  3. Mar 15, 2015 #2
    I think you are right. The step only "divides both sides of the equation by the quantity in the large brackets on the left side" as the text states.

    So, the sign doesn't change and will be the sum of the two terms in the denominator as you wrote out above.
     
  4. Mar 15, 2015 #3
    But the the mistake is a few steps before that where the text reads "Multiplying out the last term on the right". The writer removes the brackets but only changes the sign for the first term of the brackets. It is supposed to be a minus sign in the denominator (as in a difference of the two terms).
     
  5. Mar 15, 2015 #4

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    Thank you, TitasB!
     
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