\(Y\) is a random variable with mean \(\mu=2\) and variance \(\sigma^{2}=25\).

a. Suppose you know the value of \(\mu\).

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 \(\mu\) but you have access to a random sample of size \(n=10\) from the same population. Let \(\bar{Y}\) denote the sample mean from this random sample. You predict the value of \(Y\) using \(\bar{Y}\).

i. Show that the prediction error can be decomposed as \(Y-\bar{Y}=(Y-\mu)-\) \((\bar{Y}-\mu)\), where \((Y-\mu)\) is the prediction error of the oracle predictor and \((\mu-\bar{Y})\) is the error associated with using \(\bar{Y}\) as an estimate of \(\mu\).

ii. Show that \((Y-\mu)\) has a mean of 0 , that \((\bar{Y}-\mu)\) has a mean of 0 , and that \(Y-\bar{Y}\) has a mean of 0 .

iii. Show that \((Y-\mu)\) and \((\bar{Y}-\mu)\) are uncorrelated.

iv. Show that the MSPE of \(\bar{Y}\) is MSPE \(=E(Y-\mu)^{2}+E(\bar{Y}-\mu)^{2}=\) \(\operatorname{var}(Y)+\operatorname{var}(\bar{Y})\).

v. Show that MSPE \(=25(1+1 / 10)=27.5\).