- The discrete random variable \(X\) has probability distribution
| \(x\) | - 3 | - 1 | 1 | 2 | 4 |
| \(\mathrm { P } ( X = x )\) | \(q\) | \(\frac { 7 } { 30 }\) | \(\frac { 7 } { 30 }\) | \(q\) | \(r\) |
where \(q\) and \(r\) are probabilities.
- Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
- Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\)
Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
- find the value of \(q\) and the value of \(r\)
- Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)