Sum of independent uniforms

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 \([ 0,5 ]\).

1. 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
   \(\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 25 } x & 0 \leqslant x \leqslant 5 ,
   \frac { 1 } { 25 } ( 10 - x ) & 5 < x \leqslant 10 ,
   0 & \text { otherwise } . \end{cases}\)
2. 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}
   \end{table}
3. Use the simulation to estimate \(\mathrm { P } ( T > 18 )\).
4. 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 \(T\).
5. 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.
6. Find the probability that the total waiting time for Jonathan in a year is more than 510 minutes.
   [0pt] [3]

10 Ben takes an underground train to work and back home each day. The waiting time is defined as the time from when he reaches the station platform until he boards the train. On his way to work the waiting time is \(X\) minutes, where \(X\) is modelled by a continuous uniform distribution on \([ 0,6 ]\). On his way back from work, the waiting time is \(Y\) minutes, where \(Y\) is modelled by a continuous uniform distribution on [0,4]. Ben's total waiting time for both journeys is \(Z\) minutes, where \(Z = X + Y\). You should assume that \(X\) and \(Y\) are independent.

1. Find \(\mathrm { E } ( \mathrm { Z } )\).
2. Ben thinks that \(Z\) will be well modelled by a continuous uniform distribution on \([ 0,10 ]\). By considering variances, show that he is not correct.
3. Ben's friend Jamila constructs the spreadsheet below, which shows a simulation of 20 values of \(X , Y\) and \(Z\). All of the values have been rounded to 2 decimal places.

   \multirow[b]{3}{*}{ 1 2 }	A	B	C
   	X	Y	Z
   	1.17	3.83	5.01
   3	2.01	0.81	2.82
   4	1.27	1.52	2.78
   5	1.41	3.94	5.35
   6	4.11	2.94	7.05
   7	1.76	0.96	2.72
   8	3.29	0.98	4.27
   9	0.77	0.22	0.99
   10	0.99	1.44	2.43
   11	4.79	2.43	7.22
   12	3.82	3.93	7.75
   13	5.25	2.74	7.99
   14	2.64	0.48	3.12
   15	1.54	2.18	3.72
   16	2.71	1.66	4.36
   17	0.04	3.24	3.28
   18	5.95	3.12	9.07
   19	5.22	1.21	6.42
   20	4.16	0.11	4.27
   21	1.02	0.99	2.01
   22

   Write down an estimate of \(\mathrm { P } ( Z > 6 )\).
4. Use a Normal approximation to determine the probability that Ben's total waiting time when travelling to and from work on 40 days is more than 210 minutes.

10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals \([ 0,10 ]\) and \([ 0,6 ]\) respectively. The times in minutes for the bus journeys in the morning and evening are modelled by \(\mathrm { N } ( 25,4 )\) and \(\mathrm { N } ( 28,16 )\) respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable \(T\). The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of \(T\) that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of \(T\) which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.

1. Use the spreadsheet output to estimate each of the following.
   • \(\mathrm { P } ( T \leqslant 56 )\)
   • \(\mathrm { P } ( T > 61 )\)
   • The random variable \(W\) is Normally distributed with the same mean and variance as \(T\). Find each of the following.
   • \(\mathrm { P } ( W \leqslant 56 )\)
   • \(\mathrm { P } ( W > 61 )\)
   • Explain why, if many more journeys were used in the simulation, you would expect \(\mathrm { P } ( T > 61 )\) to be extremely close to \(\mathrm { P } ( W > 61 )\).

6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6

1. Find \(\mathrm { P } ( X > 4 )\) 6
2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
   6
3. 1. Find the mean total score of the game.
      6
4. (ii) Find the variance of the total score of the game.