4. A discrete random variable \(X\) has the probability distribution given in the table below, where \(a\) and \(b\) are constants.
| \(x\) | - 1 | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(\frac { 1 } { 10 }\) | \(\frac { 1 } { 5 }\) | \(\frac { 3 } { 10 }\) | \(b\) |
Given \(\mathrm { E } ( X ) = \frac { 9 } { 5 }\)
- find two simultaneous equations for \(a\) and \(b\),
- show that \(a = \frac { 1 } { 20 }\) and find the value of \(b\).
- Specify the cumulative distribution function \(\mathrm { F } ( x )\) for \(x = - 1,0,1\), 2 and 3
- Find \(\mathrm { P } ( X < 2.5 )\).
- Find \(\operatorname { Var } ( 3 - 2 X )\).
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