4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).
- Write down the mean of \(X\).
- Find \(\mathrm { P } ( X \leqslant 2.4 )\)
- Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\)
The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
- Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
- Find \(\operatorname { Var } ( Y )\).
- Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).