5.02i Poisson distribution: random events model

2 The number of bacteria in 1 ml of drug \(A\) has a Poisson distribution with mean 0.5. The number of the same bacteria in 1 ml of drug \(B\) has a Poisson distribution with mean 0.75 . A mixture of these drugs used to treat a particular disease consists of 1.4 ml of drug \(A\) and 1.2 ml of drug \(B\). Bacteria in the drugs will cause infection in a patient if 5 or more bacteria are injected.

1. Calculate the probability that, in a sample of 20 patients treated with the mixture, infection will occur in no more than one patient.
2. State an assumption required for the validity of the calculation.

1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.

1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
2. State one assumption needed in the calculation.

1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.

2 Scientists researching into the chemical composition of dust in space collect specimens using a specially designed spacecraft. The craft collects the particles of dust in trays that are made up of a large array of cells containing aerogel. The aerogel traps the particles that penetrate into the cells.

1. For a random sample of 100 cells, the number of particles of dust in each cell was counted, giving the following results.

   Number of particles	0	1	2	3	4	5	6	7	8	9	\(10 +\)
   Frequency	4	7	10	20	17	15	10	9	5	3	0

   It is thought that the number of particles collected in each cell can be modelled using the distribution Poisson(4.2) since 4.2 is the sample mean for these data. Some of the calculations for a \(\chi ^ { 2 }\) test are shown below. The cells for 8,9 and \(10 +\) particles have been combined.

   Number of particles

   Observed frequency

   Expected frequency

   Contribution to \(X ^ { 2 }\)

   5	6	7	\(8 +\)
   15	10	9	8
   16.33	11.44	6.86	6.39
   0.1083	0.1813	0.6676	0.4056

   Complete the calculations and carry out the test using a \(10 \%\) significance level to see whether the number of particles per cell may be modelled in this way.
2. The diameters of the dust particles are believed to be distributed symmetrically about a median of 15 micrometres \(( \mu \mathrm { m } )\). For a random sample of 20 particles, the sum of the signed ranks of the diameters of the particles smaller than \(15 \mu \mathrm {~m} \left( W _ { - } \right)\)is found to be 53 . Test at the \(5 \%\) level of significance whether the median diameter appears to be more than \(15 \mu \mathrm {~m}\).

4 The numbers of call-outs per day received by a fire station for a random sample of 255 weekdays were recorded as follows. The mean number of call-outs per day for these data is 0.6 . A Poisson model, using this sample mean of 0.6 , is fitted to the data, and gives the following expected frequencies (correct to 3 decimal places).

1. Using a \(5 \%\) significance level, carry out a test to examine the goodness of fit of the model to the data. The time \(T\), measured in days, that elapses between successive call-outs can be modelled using the exponential distribution for which \(\mathrm { f } ( t )\), the probability density function, is $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 , \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 , \end{cases}$$ where \(\lambda\) is a positive constant.
2. For the distribution above, it can be shown that \(\mathrm { E } ( T ) = \frac { 1 } { \lambda }\). Given that the mean time between successive call-outs is \(\frac { 5 } { 3 }\) days, write down the value of \(\lambda\).
3. Find \(\mathrm { F } ( t )\), the cumulative distribution function.
4. Find the probability that the time between successive call-outs is more than 1 day.
5. Find the median time that elapses between successive call-outs.

3

1. A personal trainer believes that drinking a glass of beetroot juice an hour before exercising enables endurance tests to be completed more quickly. To test his belief he takes a random sample of 12 of his trainees and, on two occasions, asks them to carry out 100 repetitions of a particular exercise as quickly as possible. Each trainee drinks a glass of water on one occasion and a glass of beetroot juice on the other occasion. The times in seconds taken by the trainees are given in the table.

   Trainee	Water	Beetroot juice
   A	75.1	72.9
   B	86.2	79.9
   C	77.3	71.6
   D	89.1	90.2
   E	67.9	68.2
   F	101.5	95.2
   G	82.5	76.5
   H	83.3	80.2
   I	102.5	99.1
   J	91.3	82.2
   K	92.5	90.1
   L	77.2	77.9

   The trainer wishes to test his belief using a paired \(t\) test at the \(1 \%\) level of significance. Assuming any necessary assumptions are valid, carry out a test of the hypotheses \(\mathrm { H } _ { 0 } : \mu _ { D } = 0 , \mathrm { H } _ { 1 } : \mu _ { D } < 0\), where \(\mu _ { D }\) is the population mean difference in times (time with beetroot juice minus time with water).
2. An ornithologist believes that the number of birds landing on the bird feeding station in her garden in a given interval of time during the morning should follow a Poisson distribution. In order to test her belief, she makes the following observations in 60 randomly chosen minutes one morning.

   Number of birds	0	1	2	3	4	5	6	\(\geqslant 7\)
   Frequency	2	5	10	17	14	7	4	1

   Given that the data in the table have a mean value of 3.3, use a goodness of fit test, with a significance level of \(5 \%\), to investigate whether the ornithologist is justified in her belief.

5 The random variable \(X\) has a Poisson distribution with mean \(\lambda\). It is given that the moment generating function of \(X\) is \(e ^ { \lambda \left( e ^ { t } - 1 \right) }\).

1. Use the moment generating function to verify that the mean of \(X\) is \(\lambda\), and to show that the variance of \(X\) is also \(\lambda\).
2. Five independent observations of \(X\) are added to produce a new variable \(Y\). Find the moment generating function of \(Y\), simplifying your answer.

1 Traffic engineers are studying the flow of vehicles along a road. At an initial stage of the investigation, they assume that the average flow remains the same throughout the working day. An automatic counter records the number of vehicles passing a certain point per minute during the working day. A random sample of these records is selected; the sample values are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\).

1. The engineers model the underlying random variable \(X\) by a Poisson distribution with unknown parameter \(\theta\). Obtain the likelihood of \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n }\) and hence find the maximum likelihood estimate of \(\theta\).
2. Write down the maximum likelihood estimate of the probability that no vehicles pass during a minute.
3. The engineers note that, in a sample of size 1000 with sample mean \(\bar { x } = 5\), there are no observations of zero. Suggest why this might cast some doubt on the investigation.
4. On checking the automatic counter, the engineers find that, due to a fault, no record at all is made if no vehicle passes in a minute. They therefore model \(X\) as a Poisson random variable, again with an unknown parameter \(\theta\), except that the value \(x = 0\) cannot occur. Show that, under this model, $$\mathrm { P } ( X = x ) = \frac { \theta ^ { x } } { \left( \mathrm { e } ^ { \theta } - 1 \right) x ! } , \quad x = 1,2 , \ldots$$ and hence show that the maximum likelihood estimate of \(\theta\) satisfies the equation $$\frac { \theta \mathrm { e } ^ { \theta } } { \mathrm { e } ^ { \theta } - 1 } = \bar { x }$$

10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by \(T\) minutes. Show that \(\mathrm { P } ( T > t ) = \mathrm { e } ^ { - 0.8 t }\) and hence show that \(T\) has a negative exponential distribution. Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.

8 A certain mechanical component has a lifetime, \(T\) months, which has a negative exponential distribution with mean 2.5.

1. A machine is fitted with 5 of these components which function independently.
   1. Find the probability that all 5 components are operating 3 months after being fitted.
   2. Find also the probability that exactly two components fail within one month of being fitted.
2. Show that the probability that \(n\) independent components are all operating \(c\) months after being fitted is equal to the probability that a single component is operating \(n c\) months after being fitted.

A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table. It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.

1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
2. Carry out a goodness of fit test at the \(10 \%\) significance level.

9 Cotton cloth is sold from long rolls of cloth. The number of flaws on a randomly chosen piece of cloth of length \(a\) metres has a Poisson distribution with mean \(0.8 a\). The random variable \(X\) is the length of cloth, in metres, between two successive flaws.

1. Explain why, for \(x \geqslant 0 , \mathrm { P } ( X > x ) = \mathrm { e } ^ { - 0.8 x }\).
2. Find the probability that there is at least one flaw in a 4 metre length of cloth.
3. Find
   1. the distribution function of \(X\),
   2. the probability density function of \(X\),
   3. the interquartile range of \(X\).

7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable \(T\). The probability density function f of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where \(\lambda\) is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .

4 The number of printers, \(V\), bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, \(W\), bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4

1. Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
   [0pt] [2 marks] 4
2. State the circumstance under which the distributional model you used in part (a) would not be valid.
   [0pt] [1 mark]

3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).

1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.

2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

5 The manager of an emergency response hotline believes that calls are made to the hotline independently and at constant average rate throughout the day. From a small random sample of the population, the manager finds that the mean number of calls made in a 1-hour period is 14.4. Let \(R\) denote the number of calls made in a randomly chosen 1-hour period.

1. Using evidence from the small sample, state a suitable distribution with which to model \(R\). You should give the value(s) of any parameter(s).
2. In this part of the question, use the distribution and value(s) of the parameter(s) from your answer to part (a).
   1. Find \(\mathrm { P } ( R > 20 )\).
   2. Given that \(\mathrm { P } ( \mathrm { R } = \mathrm { r } ) > \mathrm { P } ( \mathrm { R } = \mathrm { r } + 1 )\), show algebraically that \(r > 13.4\).
   3. Hence write down the mode of the distribution. The manager also finds, from records over many years, that the modal value of \(R\) is 10 .
3. Use this result to comment on the validity of the distribution used in part (b).
4. Assume now that the type of distribution used in part (b) is valid. Find the range(s) of values of the parameter(s) of this distribution that would correspond to the modal value of \(R\) being 10.

1 A radar device is used to detect flaws in motorway roads before they become dangerous. The number of flaws in a 1 km stretch of motorway is denoted by \(X\). It may be assumed that flaws occur randomly.

1. State two further assumptions that are necessary for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be modelled by distribution \(\operatorname { Po } ( 5.7 )\).
2. Determine the probability that in a randomly chosen stretch of motorway, of length 1 km , there are between 8 and 11 flaws, inclusive.
3. Determine the probability that in two randomly chosen, non-overlapping, stretches of motorway, each of length 5 km , there are at least 30 flaws in one stretch and fewer than 30 flaws in the other stretch.

6 A statistician investigates the number, \(F\), of signal failures per week on a railway network.

1. The statistician assumes that signal failures occur randomly. Explain what this statement means.
2. State two further assumptions needed for \(F\) to be well modelled by a Poisson distribution. In a random sample of 50 weeks, the statistician finds that the mean number of failures per week is 1.61, with standard deviation 1.28.
3. Explain whether this suggests that \(F\) is likely to be well modelled by a Poisson distribution. Assume first that \(F \sim \operatorname { Po } ( 1.61 )\).
4. Write down an exact expression for \(\mathrm { P } ( F = 0 )\).
5. Complete the table in the Printed Answer Booklet to show the probabilities of different values of \(F\), correct to three significant figures.

   Value of \(F\)	0	1	\(\geqslant 2\)
   Probability	0.200

   After further investigation, the statistician decides to use a different model for the distribution of \(F\). In this model it is now assumed that \(\mathrm { P } ( F = 0 )\) is still 0.200 , but that if one failure occurs, there is an increased probability that further failures occur.
6. Explain the effect of this assumption on the value of \(\mathrm { P } ( F = 1 )\).

8

1. A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
2. The random variable \(X\) represents the number of particles emitted by a substance in a fixed time interval \(t\) minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for \(X\) if the value of \(t\) is very large.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = \mathrm { P } ( \mathrm { Y } = \mathrm { r } + 1 )\) \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = 1.5 \times \mathrm { P } ( \mathrm { Y } = \mathrm { r } - 1 )\). Determine the following, in either order.
   \section*{END OF QUESTION PAPER}

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.

4 The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.

1. State a further assumption needed for the Poisson model to be valid.
2. State the value of the standard deviation of \(R\).
3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\operatorname { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.

8 A team of researchers have reason to believe that the number of calls received in randomly chosen 10-minute intervals to a call centre can be well modelled by a Poisson distribution. To test this belief the researchers record the number of telephone calls received in 60 randomly chosen 10-minute intervals. The results, together with relevant calculations, are shown in the following table.

1. Calculate the mean of the observed number of calls received.
2. Calculate the variance of the observed number of calls received.
3. Comment on what your answers to parts (a) and (b) suggest about the proposed model.
4. Explain why it is necessary to combine some cells in the table.
5. Show how the values 15.506 and 0.793 in the table were obtained.
6. Carry out the test, at the \(5 \%\) significance level. In the light of the result of the test, the team consider that a different model is appropriate. They propose the following improved model: $$P ( R = r ) = \begin{cases} \frac { 1 } { 60 } ( a + ( 2 - r ) b ) & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are integers.
7. Use at least three of the observed frequencies to suggest appropriate values for \(a\) and \(b\). You should consider more than one possible pair of values, and explain which pair of values you consider best. (Do not carry out a goodness-of-fit test.)

5 Some bird-watchers study the song of chaffinches in a particular wood. They investigate whether the number, \(N\), of separate bursts of song in a 5 minute period can be modelled by a Poisson distribution. They assume that a burst of song can be considered as a single event, and that bursts of song occur randomly. \section*{(a) State two further assumptions needed for \(N\) to be well modelled by a Poisson distribution.} The bird-watchers record the value of \(N\) in each of 60 periods of 5 minutes. The mean and variance of the results are 3.55 and 5.6475 respectively.
(b) Explain what this suggests about the validity of a Poisson distribution as a model in this context. The complete results are shown in the table. The bird-watchers carry out a \(\chi ^ { 2 }\) goodness of fit test at the \(5 \%\) significance level.
(c) State suitable hypotheses for the test.
(d) Determine the contribution to the test statistic for \(n = 3\).
(e) The total value of the test statistic, obtained by combining the cells for \(n \leqslant 1\) and also for \(n \geqslant 6\), is 9.202 , correct to 4 significant figures. Complete the goodness of fit test.
(f) It is known that chaffinches are more likely to sing in the presence of other chaffinches. Explain whether this fact affects the validity of a Poisson model for \(N\).

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
2. (a) Write down an expression for \(\mathrm { P } ( X = r )\).
   (b) Hence find \(\mathrm { P } ( X = 3 )\).
3. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.