1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula
\(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).
- Complete the copy of the table in the Printed Answer Booklet.
| \(r\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | \(91 k\) | \(61 k\) | \(37 k\) | | | |
- Show that \(k = \frac { 1 } { 216 }\).
- Draw a graph to illustrate the distribution.
- Comment briefly on the shape of the distribution.
- In this question you must show detailed reasoning.
Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)