5. The score when a spinner is spun is given by the discrete random variable \(X\) with the following probability distribution, where \(a\) and \(b\) are probabilities.
| \(x\) | - 1 | 0 | 2 | 4 | 5 |
| \(\mathrm { P } ( X = x )\) | \(b\) | \(a\) | \(a\) | \(a\) | \(b\) |
- Explain why \(\mathrm { E } ( X ) = 2\)
- Find a linear equation in \(a\) and \(b\).
Given that \(\operatorname { Var } ( X ) = 7.1\)
- find a second equation in \(a\) and \(b\) and simplify your answer.
- Solve your two equations to find the value of \(a\) and the value of \(b\).
The discrete random variable \(Y = 10 - 3 X\)
- Find
- \(\mathrm { E } ( Y )\)
- \(\operatorname { Var } ( Y )\)
The spinner is spun once.
- Find \(\mathrm { P } ( Y > X )\).