Waiting time applications

1. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable \(X\) with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 2 \leqslant x \leqslant 7
0 & \text { otherwise } \end{cases}$$

1. Write down the name of this distribution. A randomly selected flight takes off at 9am
2. Find the probability that the next flight takes off before 9.05 am
3. Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes.
4. Find the cumulative distribution function of \(X\), for all \(x\)
5. Sketch the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 7\) On foggy days, an extra 2 minutes is added to each waiting time.
6. Find the mean and variance of the waiting times between flight take-offs on foggy days.

1. Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15

• It always takes Navtej 3 minutes to walk to the bus stop
• Buses run every 15 minutes and Navtej catches the first bus that arrives
• Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work

The total time, \(T\) minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval \([ \alpha , \beta ]\)

1. 1. Show that \(\alpha = 32\)
   2. Show that \(\beta = 47\)
2. State fully the probability density function for this distribution.
3. Find the value of
   1. \(\mathrm { E } ( T )\)
   2. \(\operatorname { Var } ( T )\)
4. Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.

4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
2. Calculate the mean time of arrival of the bus.
3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
4. Find the probability that Jean is late for work.

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5
   4 - 4 t , & 0.5 \leq t \leq 1 ,
   0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
   (4) END

3 Mehreen lives a 2-minute walk away from a tram stop. Trams run every 10 minutes into the city centre, taking 20 minutes to get there. Every morning, Mehreen leaves her house, walks to the tram stop and catches the first tram that arrives. When she arrives at the city centre, she then has a 5-minute walk to her office. The total time, \(T\) minutes, for Mehreen's journey from house to office may be modelled by a rectangular distribution with probability density function $$\mathrm { f } ( t ) = \begin{cases} 0.1 & a \leqslant t \leqslant b
0 & \text { otherwise } \end{cases}$$

1. 1. Explain why \(a = 27\).
   2. State the value of \(b\).
2. Find the values of \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Find the probability that the time for Mehreen's journey is within 5 minutes of half an hour.