Ordinary Least Squares Univariate Proof

The model is constructed like:

This is estimated as:

As a starting point, the regression line must pass through these two points:

and

Take the generic equation form of a line:

Insert the first point into this form.

This can be trivially rewritten to solve for a in terms of b:

Insert the second point into the original form.

Now additionally insert the solution for a in terms of b.

Expand all terms to produce:

This can now be eliminated into:

Giving a solution for b:

This solution is trivially rewritten as:

Expand the formula for correlation as:

This can now be eliminated into:

Finally, b can be eloquently written as:

Giving a generic formula for the regression line:

Statistics/OrdinaryLeastSquares/Univariate