Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / M / 3 / c / ii

The identity $Y=aX+W$ forces $b=1-2a$ and $c=1-a$. Independence will follow from zero covariance, because $X,W$ are jointly Gaussian random variables. That covariance isThusThese values give $W=(-Z_1+2Z_2)/5$ with variance $1/5$ and zero covariance with $X$. Joint normality then proves the requested independence.