I am studying the best linear predictor part of Conditional Expectation and the Projection part about the regression model.

while we are looking for the Beta that minimizes S(B), I quite do not understand how the first function can be written in the second one:

S(B) = E[(Y-X'B)^2] - first S(B) = E[Y^2] -2B'E[XY]+B'E[XX']B -second

' refers to transpose. For example, I dont understand why we have B' and X in the second one which was absent in the first one. I guess there is some manipulation or traits of expectation that I am missing.



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