2. The discrete random variable \(X\) has the following probability distribution, where \(p\) and \(q\) are constants.
| \(x\) | - 2 | - 1 | \(\frac { 1 } { 2 }\) | \(\frac { 3 } { 2 }\) | 2 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(q\) | 0.2 | 0.3 | \(p\) |
- Write down an equation in \(p\) and \(q\)
Given that \(\mathrm { E } ( X ) = 0.4\)
- find the value of \(q\)
- hence find the value of \(p\)
Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.275\)
- find \(\operatorname { Var } ( X )\)
Sarah and Rebecca play a game.
A computer selects a single value of \(X\) using the probability distribution above.
Sarah's score is given by the random variable \(S = X\) and Rebecca's score is given by the random variable \(R = \frac { 1 } { X }\) - Find \(\mathrm { E } ( R )\)
Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.
- Find the probability that
- Sarah is the winner,
- Rebecca is the winner.