9 Every weekday Jonathan takes an underground train to work. On any weekday the time in minutes that he has to wait at the station for a train is modelled by the continuous uniform distribution over \([ 0,5 ]\).
- Find the probability that Jonathan has to wait at least 3 minutes for a train.
The total time that Jonathan has to wait on two days is modelled by the continuous random variable \(X\) with probability density function given by
\(\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 25 } x & 0 \leqslant x \leqslant 5 ,
\frac { 1 } { 25 } ( 10 - x ) & 5 < x \leqslant 10 ,
0 & \text { otherwise } . \end{cases}\) - Find the probability that Jonathan has to wait a total of at most 6 minutes on two days.
Jonathan's friend suggests that the total waiting time for 5 days, \(T\) minutes, will almost certainly be less than 18 minutes. In order to investigate this suggestion, Jonathan constructs the simulation shown in Fig. 9. All of the numbers in the simulation have been rounded to 2 decimal places.
\begin{table}[h]
| A | B | C | D | E | F |
| 1 | Mon | Tue | Wed | Thu | Fri | Total T |
| 2 | 1.78 | 4.36 | 2.74 | 3.88 | 4.64 | 17.41 |
| 3 | 0.95 | 1.30 | 4.83 | 4.29 | 1.81 | 13.18 |
| 4 | 4.27 | 4.90 | 4.57 | 1.41 | 3.66 | 18.81 |
| 5 | 0.80 | 0.06 | 3.20 | 1.76 | 0.35 | 6.17 |
| 6 | 0.03 | 4.82 | 1.26 | 3.53 | 0.13 | 9.77 |
| 7 | 3.88 | 4.73 | 1.19 | 3.75 | 1.29 | 14.84 |
| 8 | 4.11 | 3.54 | 4.33 | 0.77 | 4.50 | 17.25 |
| 9 | 3.54 | 0.11 | 3.85 | 2.86 | 1.58 | 11.94 |
| 10 | 1.87 | 1.82 | 3.00 | 3.53 | 1.83 | 12.05 |
| 11 | 4.00 | 2.98 | 4.59 | 1.73 | 1.76 | 15.06 |
| 12 | 1.91 | 3.85 | 2.08 | 1.72 | 2.82 | 12.38 |
| 13 | 0.10 | 4.86 | 2.51 | 0.52 | 2.17 | 10.15 |
| 14 | 1.24 | 4.26 | 0.95 | 1.33 | 1.78 | 9.57 |
| 15 | 2.99 | 0.69 | 3.85 | 3.41 | 2.42 | 13.36 |
| 16 | 4.67 | 1.76 | 2.13 | 3.48 | 3.10 | 15.14 |
| 17 | 1.94 | 1.07 | 0.91 | 0.63 | 3.34 | 7.89 |
| 18 | 0.11 | 2.29 | 0.71 | 4.21 | 0.86 | 8.18 |
| 19 | 0.43 | 4.58 | 4.89 | 1.86 | 2.84 | 14.60 |
| 20 | 4.23 | 0.88 | 2.71 | 4.88 | 4.20 | 16.91 |
| 21 | 3.72 | 4.58 | 3.11 | 4.89 | 3.18 | 19.49 |
\captionsetup{labelformat=empty}
\caption{Fig. 9}
- Use the simulation to estimate \(\mathrm { P } ( T > 18 )\).
- Explain how Jonathan could obtain a better estimate.
Jonathan thinks that he can use the Central Limit Theorem to provide a very good approximation to the distribution of \(T\).
- Find each of the following.
- \(\mathrm { E } ( T )\)
- \(\operatorname { Var } ( T )\)
- Use the Central Limit Theorem to estimate \(\mathrm { P } ( T > 18 )\).
- Comment briefly on the use of the Central Limit Theorem in this case.
Jonathan travels to work on 200 days in a year. - Find the probability that the total waiting time for Jonathan in a year is more than 510 minutes.
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