N2008 P2 Q11

The random variable \(X\) has the distribution \(\mathrm N(50,8^2)\). Given that \(X_1\) and \(X_2\) are two independent observations of \(X\), find

1. \(\mathrm P(X_1+X_2>120)\),[2]
2. \(\mathrm P(X_1>X_2+15)\).[3]

The random variable \(Y\) is related to \(X\) by the formula \(Y=aX+b\), where \(a\) and \(b\) are constants with \(a>0\). Given that \(\mathrm P(Y<74)=\mathrm P(Y>146)=0.0668\), find the values of \(\mathrm E(Y)\) and \(\mathrm{Var}(Y)\), and hence find the values of \(a\) and \(b\).[7]