5.02n Sum of Poisson variables: is Poisson

1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .

1. 1. Calculate \(\mathrm { P } ( X = 2 )\).
   2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
2. 1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
   2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

2 The number of computers, \(A\), bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, \(B\), bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .

1. 1. Calculate \(\mathrm { P } ( A = 4 )\).
   2. Determine \(\mathrm { P } ( B \leqslant 6 )\).
   3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
2. Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
3. The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
   1. Calculate the mean and variance of these data.
   2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.

2 A new information technology centre is advertising places on its one-week residential computer courses.

1. The number of places, \(X\), booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
   1. State the standard deviation of \(X\).
   2. Calculate \(\mathrm { P } ( 6 < X < 12 )\).
2. The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
   1. Write down the distribution for \(T\), the total number of places booked each week on the publishing and web design courses.
   2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
3. The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution \(\mathrm { Po } ( 12.0 )\) could provide a suitable model for the number of places booked each week on the database course.

5

1. In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random. Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
2. 1. In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people. A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
   2. In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people. A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
   3. The random variable \(T\) denotes the total number of people in the two selected towns who have this blood disorder. Write down the distribution of \(T\) and hence determine \(\mathrm { P } ( T > 16 )\).

3 Lucy is the captain of her school's cricket team.
The number of catches, \(X\), taken by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.6 . The number of run-outs, \(Y\), effected by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.15 .

1. Find:
   1. \(\mathrm { P } ( X \leqslant 1 )\);
   2. \(\mathrm { P } ( X \leqslant 1\) and \(Y \geqslant 1 )\).
2. State the assumption that you made in answering part (a)(ii).
3. During a particular season, Lucy plays in 16 cricket matches.
   1. Calculate the probability that the number of catches taken by Lucy during this season is exactly 10 .
   2. Determine the probability that the total number of catches taken and run-outs effected by Lucy during this season is at least 15 .

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)

3 A large office block is busy during the five weekdays, Monday to Friday, and less busy during the two weekend days, Saturday and Sunday. The block is illuminated by fluorescent light tubes which frequently fail and must be replaced with new tubes by John, the caretaker. The number of fluorescent tubes that fail on a particular weekday can be modelled by a Poisson distribution with mean 1.5. The number of fluorescent tubes that fail on a particular weekend day can be modelled by a Poisson distribution with mean 0.5 .

1. Find the probability that:
   1. on one particular Monday, exactly 3 fluorescent light tubes fail;
   2. during the two days of a weekend, more than 1 fluorescent light tube fails;
   3. during a complete seven-day week, fewer than 10 fluorescent light tubes fail.
2. John keeps a supply of new fluorescent light tubes. More new tubes are delivered every Monday morning to replace those that he has used during the previous week. John wants the probability that he runs out of new tubes before the next Monday morning to be less than 1 per cent. Find the minimum number of new tubes that he should have available on a Monday morning.
3. Give a reason why a Poisson distribution with mean 0.375 is unlikely to provide a satisfactory model for the number of fluorescent light tubes that fail between 1 am and 7 am on a weekday.

1 The number of cars, \(X\), passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.

1. Calculate \(\mathrm { P } ( X = 2 )\).
2. 1. Write down the distribution of \(Y\), the number of cars passing along this road in a 5-minute interval.
   2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.

1 The number of A-grades, \(X\), achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

1. Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
2. The number of A-grades, \(Y\), achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
   1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
   2. What assumption did you make in answering part (b)(i)?

2

1. The number of telephone calls, \(X\), received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine \(\mathrm { P } ( X = 8 )\).
2. The number of telephone calls, \(Y\), received per hour for Dr Bracken may be modelled by a Poisson distribution with mean \(\lambda\) and standard deviation 3 .
   1. Write down the value of \(\lambda\).
   2. Determine \(\mathrm { P } ( Y > \lambda )\).
3. 1. Assuming that \(X\) and \(Y\) are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
   2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
   3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.

5

1. The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
   1. at least 9 minor accidents;
   2. more than 5 but fewer than 10 minor accidents.
2. The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
3. The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
   1. state the value of \(\lambda\);
   2. determine \(\mathrm { P } ( T > 16 )\);
   3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, \(X\), within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of \(X\) can be modelled by a Poisson distribution with \(\lambda = 3.2\).

1. Use the summarised data to support Peter's belief.
2. Using Peter's model, calculate the probability that:
   1. a single placing of the quadrat contains more than 5 quartz pebbles;
   2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
   3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
3. Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
   [0pt] [3 marks]

1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:

1. a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
2. a 20 m length of tideline along this beach contains exactly 2 drinks cans;
3. a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
   [0pt] [4 marks]

1 Over a period of time, radioactive substances decay into other substances. During this decay a Geiger counter can be used to detect the number of radioactive particles that the substance emits. A certain radioactive source is decaying at a constant average rate of 6.1 particles per 10 seconds. The particles are emitted randomly and independently of each other.

1. State a distribution which can be used to model the number of particles emitted by the source in a 10-second period.
2. State the variance of this distribution.
3. Find the probability that at least 6 particles are detected in a period of 10 seconds.
4. Find the probability that at least 36 particles are detected in a period of 60 seconds.
5. Another radioactive source emits particles randomly and independently at a constant average rate of 1.7 particles per 5 seconds. Find the probability that at least 10 but no more than 15 particles are detected altogether from the two sources in a period of 10 seconds.

8 The time in hours to failure of a component may be modelled by an exponential distribution with parameter \(\lambda = 0.025\) In a manufacturing process, the machine involved uses one of these components continuously until it fails. The component is then immediately replaced.
8

1. Write down the mean time to failure for a component. 8
2. Find the probability that a component will fail during a 12-hour shift. 8
3. A component has not failed for 30 hours. Find the probability that this component lasts for at least another 30 hours.
   [0pt] [2 marks] 8
4. Find the probability that a component does not fail during 4 consecutive 12-hour shifts.
   [0pt] [3 marks]
   8
5. (i) State the distribution that can be used to model the number of components that fail during one hour of the manufacturing process.
   [0pt] [2 marks]
   8 (e) (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
   [0pt] [2 marks]

The number of heaters, \(H\), bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7

1. Calculate \(\mathrm { P } ( H \geqslant 2 )\) The number of heaters, \(G\), bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where \(G\) and \(H\) are independent.
2. Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places.
3. Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days. December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14
4. Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a \(5 \%\) level of significance and state your hypotheses clearly.

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1. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8

The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3

1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
   State your hypotheses clearly.

Rowan and Alex are both check-in assistants for the same airline. The number of passengers, \(R\), checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28

1. Calculate \(\mathrm { P } ( R \geqslant 23 )\) The number of passengers, \(A\), checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where \(R\) and \(A\) are independent. A randomly selected 30-minute period is chosen.
2. Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods.
3. Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working.

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows

$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$

1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)

1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   1. \(\mathrm { P } ( X + Y ) > 40\),
   2. \(\operatorname { Var } ( 2 X - Y )\).
   3. State a necessary assumption for your calculations in part (i) to be valid.

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

1. Calculate:
   1. \(\mathrm { P } ( X \leqslant 3 )\);
   2. \(\quad \mathrm { P } ( Y = 5 )\).
2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
   1. Write down the distribution of \(T\), stating any assumption that you make.
   2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
   3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
   4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
   1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
   2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.