4. Let X, Y be two random variables. We want to investigate if there is a linear relationship between them. If there is, we should be able to approximate Y witlh a + bX for some constants a,b. To evaluate the fit of such approximation, we measure the error by considering the average squared distance between Y and a + bX, that is, expanding the RHS A small E means a good fit . (a) Determine the critical values of a, b by setting the partial derivatives (a, b) (b) With the critical values from parta), the minimum value of E(a, ó) (after lots of cancellation) is (1-r2 y) Var(y), where ρxy s the correlation coefficient between X, Y. With this conclusion, determine the range of ρχ.y

Question
4. Let X, Y be two random variables. We want to investigate if there is a linear
relationship between them. If there is, we should be able to approximate Y witlh
a + bX for some constants a,b. To evaluate the fit of such approximation, we
measure the error by considering the average squared distance between Y and
a + bX, that is,
expanding the RHS
A small E
means a good fit
.
(a) Determine the critical values of a, b by setting the partial derivatives
(a, b)
(b) With the critical values from parta), the minimum value of E(a, ó) (after lots
of cancellation) is (1-r2 y) Var(y), where ρxy s the correlation coefficient
between X, Y. With this conclusion, determine the range of ρχ.y

Image Transcription

4. Let X, Y be two random variables. We want to investigate if there is a linear relationship between them. If there is, we should be able to approximate Y witlh a + bX for some constants a,b. To evaluate the fit of such approximation, we measure the error by considering the average squared distance between Y and a + bX, that is, expanding the RHS A small E means a good fit . (a) Determine the critical values of a, b by setting the partial derivatives (a, b) (b) With the critical values from parta), the minimum value of E(a, ó) (after lots of cancellation) is (1-r2 y) Var(y), where ρxy s the correlation coefficient between X, Y. With this conclusion, determine the range of ρχ.y

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