Let \(X\) and \(Y\) be random variables with expected values \(\mu=\mu_{X}=\mu_{Y}\) and variances \(\sigma^{2}=\sigma_{X}^{2}=\sigma_{Y}^{2}\). Let \(Z=(2 X+Y) / 2\).

a. Find the expected value of \(Z\).

b. Find the variance of \(Z\) assuming \(X\) and \(Y\) are statistically independent.

c. Find the variance of \(Z\) assuming that the correlation between \(X\) and \(Y\) is -0.5 .

d. Let the correlation between \(X\) and \(Y\) be -0.5 . Find the correlation between \(a X\) and \(b Y\), where \(a\) and \(b\) are any nonzero constants.