5.02m Poisson: mean = variance = lambda

1 The number of failures of a machine each week at a factory is modelled by a Poisson distribution with mean 0.45.

1. Write down the variance of the distribution.
2. Find the probability that there are exactly 2 failures in a week.
3. State a distribution which can be used to model the number of failures in a period of 4 weeks.
4. Find the probability that there are at least 2 failures in a period of 4 weeks.

4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of \(10 \mathrm {~m} ^ { 2 }\) over a period of months.

1. Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
   Fig. 4 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D
   1	Number of coins found	Observed frequency	Expected frequency	Chi-squared contribution
   2	0	13	13.8069	0.0472
   3	1	28
   4	2	30	27.0643	0.3184
   5	3	14	17.8625	0.8352
   6	4	10	8.8419	0.1517
   7	\(\geqslant 5\)	5		0.0015

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{table}
2. Showing your calculations, find the missing values in each of the following cells.
   For the rest of this question, you should assume that the number of coins that Zara finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution with mean 1.98.
   Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per \(10 \mathrm {~m} ^ { 2 }\) area is modelled by a Poisson distribution with mean 0.42 .
3. Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of \(10 \mathrm {~m} ^ { 2 }\).
4. Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of \(100 \mathrm {~m} ^ { 2 }\).

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h] \end{table}

1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D	E
   \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	\(\leqslant 2\)	7	0.1165	6.9887	0.0000
   3	3	5		8.0874	1.1786
   4	4	12			0.2765
   5	5	10			0.0255
   6	6	10	0.1490	8.9400	0.1257
   7	7	11	0.1086	6.5134	3.0904
   8	\(\geqslant 8\)	5	0.1440	8.6414
   9

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the hypothesis test at the 5\% significance level.
3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h] \end{table}

1. Use the results to find an estimate for \(\mu\). The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2}

   \multirow[b]{2}{*}{1}	A	B	C	D	E
   	Number of vehicles	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	36	0.2725	27.2532	2.8073
   3	1	33	0.3543	35.4291
   4	2	14			3.5400
   5	\(\geqslant 3\)	17			0.5145
   6

   \end{table}
2. Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level.
3. Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.

5 Over a long period of time, it is found that the mean number of mistakes made by a certain player when playing a particular piece of music is 5 . The number of mistakes that the player makes when playing the piece is denoted by the random variable \(Y\).

1. State two assumptions necessary for \(Y\) to be modelled by a Poisson distribution. For the remainder of this question you may assume that \(Y\) can be modelled by a Poisson distribution.
2. 1. Find the probability that the player makes exactly 3 mistakes when playing the piece.
   2. Find the probability that the player makes fewer than 3 mistakes when playing the piece.
   3. Find the probability that the player makes fewer than 6 mistakes in total when playing the piece twice, assuming that the performances are independent. In a recording studio, the player plays the piece once in the morning and once in the afternoon each day for one week (7 days). It can be assumed that all the performances are independent of each other. The performances are recorded onto two CDs, one for each of two critics, A and B, to review. The critics are interested in the total number of mistakes made by the player per day. Unfortunately, there is a recording error in one of the CDs; on this CD, every piece that is supposed to be an afternoon recording is in fact just a repeat of that morning's recording. The random variables \(M _ { 1 }\) and \(M _ { 2 }\) represent the total number of mistakes per day for the correctly recorded CD and for the wrongly recorded CD respectively.
3. By considering the values of \(\mathrm { E } \left( M _ { 1 } \right)\) and \(\mathrm { E } \left( M _ { 2 } \right)\) explain why it is not possible to use the mean number of mistakes per day on the CDs to determine which critic received the wrongly recorded CD. Each critic counts the total number of mistakes made per day, for each of the 7 days of recordings on their CD. Summary data for this is given below. Critic A: \(\quad n = 7 , \quad \sum x _ { A } = 70 , \quad \sum x _ { A } ^ { 2 } = 812\) Critic B: \(\quad \mathrm { n } = 7 , \sum \mathrm { x } _ { \mathrm { B } } = 72 , \sum \mathrm { x } _ { \mathrm { B } } ^ { 2 } = 800\)
4. By considering the values of \(\operatorname { Var } \left( M _ { 1 } \right)\) and \(\operatorname { Var } \left( M _ { 2 } \right)\) determine which critic is likely to have received the wrongly recorded CD.

2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
   For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).

5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.

1. 1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
   2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
3. Determine \(\mathrm { P } ( X < 5 )\).

2 On average 1 in 4000 people have a particular antigen in their blood (an antigen is a molecule which may cause an adverse reaction). \begin{enumerate}[label=(\alph*)] \item

1. A random sample of 1200 people is selected. The random variable \(X\) represents the number of people in the sample who have this antigen in their blood. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use either a binomial or a Poisson distribution to calculate each of the following probabilities.

4 A radioactive source contains 1000000 nuclei of a particular radioisotope. On average 1 in 200000 of these nuclei will decay in a period of 1 second. The random variable \(X\) represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\). Use a Poisson distribution to answer parts (b) and (c).
2. Calculate each of the following probabilities.

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.

1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.

The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907

1. Find the value of \(A\) The tablecloths are sold in packets of 20
   A randomly selected packet is taken.
2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
   The random variable \(X\) represents the number of these tablecloths that have no faults.
3. Find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
   The piece of cloth has area 30 square metres and is found to have 6 faults.
5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
6. Write down the \(p\)-value for the test used in part (e).

1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.

\begin{table}[h] \end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h] \end{table}

1. State suitable hypotheses for a test to investigate Anja's belief.
2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
3. Showing your working clearly, complete Table 2
4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
   1. 1
   2. 3
5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.

1. A manager keeps a record of accidents in a canteen.

Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.

1. Give a reason why a Poisson distribution could be a suitable model in this situation.
2. Assuming that a Poisson model is suitable, find the probability of
   1. at least 3 accidents in the next month,
   2. no more than 10 accidents in a 3-month period,
   3. at least 2 months with no accidents in an 8-month period. One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents. Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
3. state
   • a property of the Poisson distribution that the manager should consider when deciding how to record this situation
   • whether the manager should record this as one or two accidents
   The manager introduces some new procedures to try and reduce the average number of accidents per month. During the following 12 months the total number of accidents is 22 The manager claims that the accident rate has been reduced.
4. Use a \(5 \%\) level of significance to carry out a suitable test to assess the manager's claim.
   You should state your hypotheses clearly and the \(p\)-value used in your test.

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}

3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.

1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.

6 The number of computers sold per day by a shop can be modelled by the random variable \(Y\) where \(Y \sim \operatorname { Po } ( 42 )\) 6

1. State the variance of \(Y\) 6
2. One month ago, the shop started selling a new model of computer.
   On a randomly chosen day in the last month, the shop sold 53 computers.
   Carry out a hypothesis test, at the \(5 \%\) level of significance, to investigate whether the mean number of computers sold per day has increased in the last month.
   [0pt] [6 marks]
   6
3. Describe, in the context of the hypothesis test in part (b), what is meant by a Type II error.

8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.

1. State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second. The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
2. Find the probability that the detector detects
   (A) no neutrons in a randomly chosen second,
   (B) at least 60 neutrons in a randomly chosen period of 1 minute. A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons. If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
3. Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
4. Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.

3 The number of signal failures in a certain region of the railway network averages 10 every 3 weeks. Assume that signal failures occur independently, randomly and at constant mean rate.

1. Find the probability that
   1. there are between 7 and 12 (inclusive) signal failures in a three-week period,
   2. there are more than 4 signal failures in a one-week period.
   3. It has been calculated, using a suitable distributional approximation, that the probability of more than 62 signal failures in a period of \(n\) weeks is 0.0385 . Find the value of \(n\).

2

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).

1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.