9 The random variable \(X\) has a discrete uniform distribution over the values \(\{ 0,1,2 , \ldots , 20 \}\).
- Find \(\mathrm { P } ( X \leqslant 7 )\).
- Find each of the following.
- \(\mathrm { E } ( X )\)
- \(\operatorname { Var } ( X )\)
The spreadsheet shows a simulation of the distribution of \(X\). Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of \(X\) together with the value of the mean of the 10 values, denoted by \(Y\).
| \includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279} | A | B | C | D | E | F | G | H | I | J | K | L |
| 1 | \(X _ { 1 }\) | \(X _ { 2 }\) | \(X _ { 3 }\) | \(X _ { 4 }\) | \(X _ { 5 }\) | \(X _ { 6 }\) | \(X _ { 7 }\) | \(X _ { 8 }\) | \(X _ { 9 }\) | \(X _ { 10 }\) | \(Y\) | |
| 2 | 1 | 6 | 2 | 1 | 18 | 6 | 4 | 9 | 11 | 11 | 6.9 | |
| 3 | 13 | 14 | 12 | 2 | 4 | 11 | 16 | 0 | 16 | 0 | 8.8 | |
| 4 | 4 | 17 | 1 | 16 | 4 | 10 | 12 | 2 | 18 | 13 | 9.7 | |
| 5 | 2 | 8 | 12 | 1 | 4 | 16 | 12 | 2 | 15 | 8 | 8.0 | |
| 6 | 7 | 15 | 16 | 0 | 4 | 7 | 1 | 13 | 0 | 20 | 8.3 | |
| 7 | 15 | 13 | 10 | 1 | 12 | 0 | 20 | 15 | 16 | 6 | 10.8 | |
| 8 | 14 | 13 | 17 | 12 | 2 | 18 | 16 | 18 | 9 | 4 | 12.3 | |
| 9 | 20 | 2 | 12 | 3 | 17 | 3 | 0 | 18 | 15 | 13 | 10.3 | |
| 10 | 2 | 12 | 5 | 12 | 2 | 6 | 0 | 9 | 10 | 15 | 7.3 | |
| 11 | 5 | 11 | 13 | 10 | 9 | 17 | 10 | 4 | 20 | 15 | 11.4 | |
| 12 | 14 | 9 | 9 | 7 | 6 | 20 | 2 | 2 | 11 | 16 | 9.6 | |
| 13 | 15 | 19 | 18 | 19 | 7 | 6 | 6 | 20 | 3 | 8 | 12.1 | |
| 14 | 5 | 10 | 6 | 4 | 1 | 19 | 15 | 8 | 17 | 18 | 10.3 | |
| 15 | 0 | 3 | 15 | 15 | 11 | 12 | 0 | 3 | 9 | 16 | 8.4 | |
| 16 | 1 | 12 | 1 | 15 | 0 | 4 | 11 | 11 | 9 | 2 | 6.6 | |
| 17 | 12 | 5 | 0 | 8 | 3 | 8 | 12 | 19 | 13 | 12 | 9.2 | |
| 18 | 9 | 5 | 1 | 13 | 5 | 4 | 18 | 1 | 1 | 19 | 7.6 | |
| 19 | 16 | 2 | 20 | 20 | 12 | 17 | 2 | 7 | 8 | 20 | 12.4 | |
| 20 | 18 | 17 | 3 | 2 | 8 | 18 | 7 | 0 | 11 | 6 | 9.0 | |
| 21 | 15 | 10 | 7 | 20 | 4 | 0 | 5 | 6 | 11 | 14 | 9.2 | |
| 22 | 3 | 9 | 10 | 14 | 2 | 1 | 8 | 6 | 0 | 7 | 6.0 | |
| 23 | 11 | 10 | 11 | 10 | 19 | 11 | 3 | 7 | 10 | 0 | 9.2 | |
| 24 | 12 | 14 | 6 | 6 | 5 | 20 | 11 | 18 | 10 | 14 | 11.6 | |
| 25 | 1 | 11 | 5 | 14 | 11 | 10 | 1 | 1 | 2 | 0 | 5.6 | |
| 26 | 0 | 14 | 7 | 11 | 18 | 5 | 10 | 20 | 11 | 9 | 10.5 | |
| 27 | | | | | | | | | | | | |
- Use the spreadsheet to estimate \(\mathrm { P } ( Y \leqslant 7 )\).
- Explain why the true value of \(\mathrm { P } ( Y \leqslant 7 )\) is less than \(\mathrm { P } ( X \leqslant 7 )\), relating your answer to \(\operatorname { Var } ( X )\) and \(\operatorname { Var } ( Y )\).
- The random variable \(W\) is the mean of 30 independent values of \(X\).
Determine an estimate of \(\mathrm { P } ( W \leqslant 7 )\).