3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants.
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.2 | 0.3 | \(b\) | \(a\) |
Given that \(\mathrm { E } ( X ) = 1.7\),
- find the value of \(a\) and the value of \(b\).
Find
- \(\mathrm { P } ( 0 < X < 1.5 )\),
- \(\mathrm { E } ( 2 X - 3 )\).
- Show that \(\operatorname { Var } ( X ) = 1.41\).
- Evaluate \(\operatorname { Var } ( 2 X - 3 )\).