2017 HCI P2 Q5

The independent random variables \(X\) and \(Y\) are normally distributed with the same mean \(7\) but different variances \(\mathrm{Var}\left( X \right)\) and \(\mathrm{Var}\left( Y \right)\), respectively. It is given that \(\mathrm{P}\left( X<10 \right)=\mathrm{P}\left( Y>6 \right)\).

1. Show that \(\mathrm{Var}\left( X \right)=9\mathrm{Var}\left( Y \right)\).[3]
2. If \(\mathrm{Var}\left( Y \right)=1\), find \(\mathrm{P}\left( X<9 \right)\).[2]