Scatter plot correlation coefficient

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kidsasd987
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Handwritten-style derivation of the correlation coefficient r = cov(x,y)/(σx σy), showing the condition r = 1 implies cov(x,y) = σx σy, followed by sum expressions over i and j with a final condition that the cross sum for i ≠ j equals 0, ending with R.H.S = L.H.S.
how can I interpret the condition "if sum of (xi-x_bar)^2(yj-y_bar)^2=0 when i=/=j?"

why does this make the line linear?
 

Discussion

Since each term is squared, it is positive definite. So for the entire sum to be equal to 0, each individual term has to be 0. Does that help?
 

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