3 A fair four-sided dice has its faces numbered \(0,1,2,3\). The dice is rolled three times. The discrete random variable \(X\) is the sum of the lowest and highest scores obtained.
- Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }\).
The table below shows the probability distribution of \(X\).
| \(r\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 1 } { 64 }\) | \(\frac { 3 } { 32 }\) | \(\frac { 13 } { 64 }\) | \(\frac { 3 } { 8 }\) | \(\frac { 13 } { 64 }\) | \(\frac { 3 } { 32 }\) | \(\frac { 1 } { 64 }\) |
- In this question you must show detailed reasoning.
Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
- The random variable \(Y\) represents the sum of 10 values of \(X\).
- State a property of the 10 values of \(X\) that would make it possible to deduce the standard deviation of \(Y\).
- Given that this property holds, determine the standard deviation of \(Y\).