8 The random variable \(X\) has a continuous uniform distribution over [0,10].
- Find the probability that, if two independent values of \(X\) are taken, one is less than 3 and the other is greater than 3 .
The random variable \(T\) denotes the sum of 5 independent values of \(X\).
- State the value of \(\mathrm { P } ( T \leqslant 25 )\).
The spreadsheet below shows the heading row and the first 20 data rows from a total of 100 data rows of a simulation of the distribution of \(X\). Each of the 100 rows shows a simulation of 5 independent values of \(X\), together with \(T\), the sum of the 5 values. All of the values have been rounded to 2 decimal places.
In column I the spreadsheet shows the number of values of \(T\) that are less than or equal to the corresponding values in column H . For example, there are 75 simulated values of \(T\) that are less than or equal to 30 .
| A | B | c | D | E | F | G | H | I |
| 1 | \(\mathrm { X } _ { 1 }\) | \(\mathrm { X } _ { 2 }\) | \(\mathrm { X } _ { 3 }\) | \(\mathrm { X } _ { 4 }\) | \(\mathrm { X } _ { 5 }\) | T | | t | Number \(\leqslant \mathrm { t }\) |
| 2 | 3.73 | 6.65 | 4.93 | 0.41 | 9.33 | 25.06 | | 0 | 0 |
| 3 | 4.95 | 6.58 | 4.48 | 2.51 | 7.26 | 25.79 | | 5 | 0 |
| 4 | 8.10 | 4.87 | 4.26 | 3.83 | 0.79 | 21.85 | | 10 | 1 |
| 5 | 6.70 | 4.10 | 5.10 | 1.82 | 6.76 | 24.48 | | 15 | 4 |
| 6 | 3.73 | 8.38 | 8.49 | 9.87 | 1.31 | 31.79 | | 20 | 23 |
| 7 | 3.22 | 4.36 | 0.12 | 1.34 | 9.49 | 18.53 | | 25 | 48 |
| 8 | 9.17 | 7.13 | 5.47 | 4.35 | 2.44 | 28.55 | | 30 | 75 |
| 9 | 3.42 | 1.93 | 6.04 | 2.99 | 8.85 | 23.24 | | 35 | 93 |
| 10 | 0.98 | 0.68 | 9.82 | 9.83 | 7.28 | 28.58 | | 40 | 99 |
| 11 | 5.86 | 1.67 | 7.77 | 4.08 | 7.14 | 26.52 | | 45 | 100 |
| 12 | 9.20 | 0.31 | 5.82 | 5.31 | 6.45 | 27.10 | | 50 | 100 |
| 13 | 7.04 | 4.30 | 2.06 | 0.06 | 4.16 | 17.62 | | | |
| 14 | 0.31 | 5.02 | 1.48 | 5.37 | 1.77 | 13.94 | | | |
| 15 | 3.77 | 6.04 | 1.21 | 7.67 | 5.01 | 23.69 | | | |
| 16 | 1.21 | 5.54 | 1.90 | 1.43 | 6.91 | 17.00 | | | |
| 17 | 9.27 | 1.98 | 5.80 | 9.37 | 9.34 | 35.76 | | | |
| 18 | 4.30 | 5.66 | 2.80 | 1.56 | 1.19 | 15.51 | | | |
| 19 | 7.15 | 3.19 | 6.89 | 5.41 | 2.18 | 24.82 | | | |
| 20 | 6.18 | 6.32 | 3.01 | 6.49 | 9.12 | 31.13 | | | |
| 21 | 5.03 | 5.99 | 5.19 | 6.97 | 3.55 | 26.73 | | | |
- Use the spreadsheet output to estimate each of the following.
- \(\mathrm { P } ( T \leqslant 25 )\)
- \(\mathrm { P } ( T > 35 )\)
- In this question you must show detailed reasoning.
The random variable \(Y\) is the mean of 100 independent values of \(T\).
Determine an estimate of \(\mathrm { P } ( Y > 26 )\).