2019 EJC P2 Q7

\({{X}_{1}}\), \({{X}_{2}}\), \({{X}_{3}}\), … are independent normally-distributed random variables with common mean \(\mu \) and different variances. For each positive integer \(n\), \(\mathrm{Var}\left( {{X}_{n}} \right)=2n\).

1. Find \(\mathrm{P}\left( \mu -1<{{X}_{2}}<\mu +1 \right)\).[3]
2. Find \(\mathrm{P}({{X}_{3}}\ge {{X}_{4}})\).[1]
3. For each \(n\), let \({{Y}_{n}}=\frac{{{X}_{1}}+{{X}_{2}}+...+{{X}_{n}}}{n}\). By finding the distribution of \({{Y}_{n}}\) in terms of \(n\), determine the smallest integer value of \(n\) such that \(\mathrm{P}\left( \mu -1<{{Y}_{n}}<\mu +1 \right)>\frac{2}{3}\).[4]