14.5 Y is a random variable with mean m = 2 and variance s2 = 25.

a. Suppose you know the value of m.

i. What is the best (lowest MSPE) prediction of the value of Y? That is, what is the oracle prediction of Y?

ii. What is the MSPE of this prediction?

b. Suppose you don’t know the value of μ but you have access to a random sample of size n = 10 from the same population. Let Y denote the sample mean from this random sample. You predict the value of Y using Y.

i. Show that the prediction error can be decomposed as Y - Y = 1Y - m2 -

1Y - μ2, where 1Y - μ2 is the prediction error of the oracle predictor and 1μ - Y2 is the error associated with using Y as an estimate of m.

ii. Show that 1Y - μ2 has a mean of 0, that 1Y - μ2 has a mean of 0, and that Y - Y has a mean of 0.

iii. Show that 1Y - μ2 and 1Y - μ2 are uncorrelated.

iv. Show that the MSPE of Y is MSPE = E1Y - m2 2 + E1Y - m2 2 =

var1Y2 + var1Y2.

v. Show that MSPE = 2511 + 1>102 = 27.5.