Three or more independent Poisson sums

7 Each day at a certain doctor's surgery there are 70 appointments available in the morning and 60 in the afternoon. All the appointments are filled every day. The probability that any patient misses a particular morning appointment is 0.04 , and the probability that any patient misses a particular afternoon appointment is 0.05 . All missed appointments are independent of each other. Use suitable approximating distributions to answer the following.

1. Find the probability that on a randomly chosen morning there are at least 3 missed appointments.
2. Find the probability that on a randomly chosen day there are a total of exactly 6 missed appointments.
3. Find the probability that in a randomly chosen 10-day period there are more than 50 missed appointments.

2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520.
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.

3. (a) State two conditions under which a Poisson distribution is a suitable model to use in statistical work. The number of cars passing an observation point in a 10 minute interval is modelled by a Poisson distribution with mean 1.
(b) Find the probability that in a randomly chosen 60 minute period there will be

1. exactly 4 cars passing the observation point,
2. at least 5 cars passing the observation point. The number of other vehicles, other than cars, passing the observation point in a 60 minute interval is modelled by a Poisson distribution with mean 12.
   (c) Find the probability that exactly 1 vehicle, of any type, passes the observation point in a 10 minute period.

4

1. A discrete random variable \(X\) has a probability function defined by $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } \quad \text { for } x = 0,1,2,3,4 , \ldots \ldots$$
   1. State the name of the distribution of \(X\).
   2. Write down, in terms of \(\lambda\), expressions for \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
   3. Write down an expression for \(\mathrm { P } ( X = x + 1 )\), and hence show that $$\mathrm { P } ( X = x + 1 ) = \frac { \lambda } { x + 1 } \mathrm { P } ( X = x )$$
2. The number of cars and the number of coaches passing a certain road junction may be modelled by independent Poisson distributions.
   1. On a winter morning, an average of 500 cars per hour and an average of 10 coaches per hour pass this junction. Determine the probability that a total of at least 10 such vehicles pass this junction during a particular 1 -minute interval on a winter morning.
   2. On a summer morning, an average of 836 cars per hour and an average of 22 coaches per hour pass this junction. Calculate the probability that a total of at most 3 such vehicles pass this junction during a particular 1 -minute interval on a summer morning. Give your answer to two significant figures.
      (3 marks)

The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520. [3]
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval. [1]