7. The random variable \(X\) follows a continuous uniform distribution over the interval [2,11].
- Write down the mean of \(X\).
- Find \(\mathrm { P } ( X \geq 8.6 )\).
- Find \(\mathrm { P } ( | X - 5 | < 2 )\).
The random variable \(Y\) follows a continuous uniform distribution over the interval \([ a , b ]\).
- Show by integration that
$$\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$$
- Hence, prove that
$$\operatorname { Var } ( Y ) = \frac { 1 } { 12 } ( b - a ) ^ { 2 }$$
You may assume that \(\mathrm { E } ( Y ) = \frac { 1 } { 2 } ( a + b )\).