Question: Take a simple linear regression model: Yi = 0 + 1xi + i , where i = 1, . . . , n and i

Take a simple linear regression model: Yi = 0 + 1xi + i , where i = 1, . . . , n and i iid N(0, 2 ). Prove that COV ( 0, 1 ) = x / sxx 2 . Hint: Write 0 and 1 in terms of ki and ci , respectively (there will be sums involved). Then find the covariance. For covariance, if a and b are constants, then COV(aX, bY ) = ab COV(X, Y ). Sums can also be pulled outside of covariance.

In class, we found that E(SSR) = 2 + ^2 1 sxx. Prove this using a different method, assuming that 1 N (1, 2 / Sxx) , where 2 is the error variance, and sxx = sigma n i=1 (xi x)^ 2 . Hint: Is there a relationship between V ( 1 ) and E ( 1 ) ?

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