Question

Instructions

1. Type or write clearly and legibly.

2. Unreadable/unclear answers will be marked wrong.

3. Explain all your logic. Do not skip steps.

1. Recall that any random variable of the form a +bZ is a linear predic tor of Y, and any such variable with a -0 is a zero-intercept linear predictor of Y (a) By studying the quadratic function E(Y -bZ)2, show that the best zero-intercept linear predictor of Y (in terms of minimizing the MSPE) is obtained by taking E(ZY) whenever the relevant expectations exist. (b) Calculate the corresponding MSPE and use it to establish the Cauchy-Schwarz inequality for Y and Z, (c) Now, consider a linear predictor a + bZ, and show that for any given b, the MSPE is minimized by a EY - bEZ (d) Substitute this a into (1) and minimize the resulting expression with respect to b using Part (a). Using your results, argue that the best linear predictor of Y is L(Z) al +01Z, where OV (e) Derive the MSPE of the best linear predictor and use it to establish the well-known correlation inequalhty,

Answer 1