2008 VJC BT P2 Q14

Two independent random variable \(X\) and \(Y\) have the distribution \(\mathrm{N}\left( 2\mu ,{{\sigma }^{2}} \right)\) and \(\mathrm{N}\left( 3\mu ,3{{\sigma }^{2}} \right)\) respectively.

1. Given that \(\mathrm{P}\left( X<2\mu +2{{\sigma }^{2}} \right)=k\), show that \(\mathrm{P}\left( Z<2\sigma \right)=k\).
2. Write down the mean and the variance of \(Y-X\). Hence find \(\mathrm{P}\left( \left| Y-X-\mu \right|>4{{\sigma }^{2}} \right)\) in terms of \(k\).