5.02k Calculate Poisson probabilities

A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.

1. Name a distribution that would be suitable for modelling the number of counts per minute. Give any parameters required for the model.
2. Find the probability of at least 4 counts in a randomly chosen minute.
3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
4. Decide whether these data support your answer to part (a).
5. Use the improved data to find probability of exactly two counts in a given one-minute interval.
   

In World War II, the number of V2 missiles that landed on each square mile of London was, on average, \(3 \cdot 5\). Assuming that the hits were randomly distributed throughout London,

1. suggest a suitable model for the number of hits on each square mile, giving a suitable value for any parameters.
2. calculate the probability that a particular square mile received
   1. no hits,
   2. more than 7 hits.
3. State, with a reason, whether the model is likely to be accurate. In contrast, the number of bombs weighing more than 1 ton landing on each square mile was 45 .
4. Use a suitable approximation to find the probability that a randomly selected square mile received more than 60 such bombs. Explain what adjustment must be made when using this approximation.
   

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results: Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2} continued ...

1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks) After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
3. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.
   

4. A hardware store is open on six days each week. On average the store sells 8 of a particular make of electric drill each week. Find the probability that the store sells

1. no more than 4 of the drills in a week,
2. more than 2 of the drills in one day. The store receives one delivery of drills at the same time each week.
3. Find the number of drills that need to be in stock after a delivery for there to be at most a 5\% chance of the store not having sufficient drills to meet demand before the next delivery.
   (3 marks)
   [0pt]

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.

1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
2. less than two visitors in a 10 -minute interval,
3. at least ten visitors in a 15-minute interval.
4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
   (5 marks)

7

1. The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
   2. Hence deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The independent Poisson random variables \(X _ { 1 }\) and \(X _ { 2 }\) are such that \(\mathrm { E } \left( X _ { 1 } \right) = 5\) and \(\mathrm { E } \left( X _ { 2 } \right) = 2\). The random variables \(D\) and \(F\) are defined by $$D = X _ { 1 } - X _ { 2 } \quad \text { and } \quad F = 2 X _ { 1 } + 10$$
   1. Determine the mean and the variance of \(D\).
   2. Determine the mean and the variance of \(F\).
   3. For each of the variables \(D\) and \(F\), give a reason why the distribution is not Poisson.
3. The daily number of black printer cartridges sold by a shop may be modelled by a Poisson distribution with a mean of 5 . Independently, the daily number of colour printer cartridges sold by the same shop may be modelled by a Poisson distribution with a mean of 2. Use a distributional approximation to estimate the probability that the total number of black and colour printer cartridges sold by the shop during a 4 -week period ( 24 days) exceeds 175.

7 The daily number of customers visiting a small arts and crafts shop may be modelled by a Poisson distribution with a mean of 24 .

1. Using a distributional approximation, estimate the probability that there was a total of at most 150 customers visiting the shop during a given 6-day period.
2. The shop offers a picture framing service. The daily number of requests, \(Y\), for this service may be assumed to have a Poisson distribution. Prior to the shop advertising this service in the local free newspaper, the mean value of \(Y\) was 2. Following the advertisement, the shop received a total of 17 requests for the service during a period of 5 days.
   1. Using a Poisson distribution, carry out a test, at the \(10 \%\) level of significance, to investigate the claim that the advertisement increased the mean daily number of requests for the shop's picture framing service.
   2. Determine the critical value of \(Y\) for your test in part (b)(i).
   3. Hence, assuming that the advertisement increased the mean value of \(Y\) to 3, determine the power of your test in part (b)(i).

7 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).

1. 1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), find, in terms of \(\lambda\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Show that \(\lambda - 1 \leqslant m \leqslant \lambda\).
   2. Given that \(\lambda = 4.9\), determine \(\mathrm { P } ( X = m )\).
3. The random variable \(Y\) has a Poisson distribution with mode \(d\) and standard deviation 15.5. Use a distributional approximation to estimate \(\mathrm { P } ( Y \geqslant d )\).
   \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-19_2484_1709_223_153}

7

1. The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
   2. Hence deduce that \(\operatorname { Var } ( X ) = \mathrm { E } ( X )\).
2. The random variable \(Y\) has a Poisson distribution with \(\mathrm { E } ( Y ) = 2.5\). Given that \(Z = 4 Y + 30\) :
   1. show that \(\operatorname { Var } ( Z ) = \mathrm { E } ( Z )\);
   2. give a reason why the distribution of \(Z\) is not Poisson.
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-19_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-20_2486_1714_221_153}

6 The demand for a WWSatNav at a superstore may be modelled by a Poisson distribution with a mean of 2.5 per day. The superstore is open 6 days each week, from Monday morning to Saturday evening.

1. 1. Determine the probability that the demand for WWSatNavs during a particular week is at most 18 .
   2. The superstore receives a delivery of WWSatNavs on each Sunday evening. The manager, Meena, requires that the probability of WWSatNavs being out of stock during a week should be at most \(5 \%\). Determine the minimum number of WWSatNavs that Meena requires to be in stock after a delivery.
2. 1. Use a distributional approximation to estimate the probability that the demand for WWSatNavs during a period of \(\mathbf { 2 }\) weeks is more than 35.
   2. Changes to the superstore's delivery schedule result in it receiving a delivery of WWSatNavs on alternate Sunday evenings. Meena now requires that the probability of WWSatNavs being out of stock during the 2 weeks following a delivery should be at most \(5 \%\). Use a distributional approximation to determine the minimum number of WWSatNavs that Meena now requires to be in stock after a delivery.
      (3 marks)

2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of \(\lambda\) per square metre. If \(\lambda\) is thought to be greater than 1.5 then action has to be taken. Using \(\mathrm { H } _ { 0 } : \lambda = 1.5\) and \(\mathrm { H } _ { 1 } : \lambda > 1.5\) a quality control officer takes a \(4 \mathrm {~m} ^ { 2 }\) sample of cloth and rejects \(\mathrm { H } _ { 0 }\) if there are 11 or more defects. If there are 8 or fewer defects she accepts \(\mathrm { H } _ { 0 }\). If there are 9 or 10 defects a second sample of \(4 \mathrm {~m} ^ { 2 }\) is taken and \(\mathrm { H } _ { 0 }\) is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.

1. Find the size of this test.
2. Find the power of this test when \(\lambda = 2\)

2. (a) Define

1. a Type I error,
2. a Type II error. Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll. The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to \(5 \%\).
   (b) Calculate the probability of a Type I error for this test.
   (c) Given that the true mean number of defects per roll of material made by the machine is now 4, calculate the probability of a Type II error.

1. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.

Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.

1. Stating your hypotheses clearly, calculate the probability of a Type I error for this test. Given that there were 9 accidents in the first year after the speed limit was increased,
2. state, giving a reason, whether or not there is evidence to support Jonty's claim.
3. Given that the value of \(\lambda\) has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased.
   

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.

2 Almost all plants of a particular species have red flowers. However on average 1 in every 1500 plants of this species have white flowers. A random sample of 2000 plants of this species is selected. The random variable \(X\) represents the number of plants in the sample that have white flowers.

1. Name two distributions which could be used to model the distribution of \(X\), stating the parameters of each of these distributions. You may use either of the distributions you have named in the rest of this question.
2. Calculate each of the following.
   Calculate the probability that there are at least 10 plants in the sample that have white flowers.

2 On a car assembly line, a robot is used for a particular task.

1. State the conditions under which a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week. It is given that the average number of breakdowns of the robot in a week is 1.7 . For the remainder of this question, you should assume that a Poisson distribution is an appropriate model for the number of breakdowns of the robot in a week.
2. 1. Find the probability that the number of breakdowns of the robot in a week is exactly 4.
   2. Determine the probability that the number of breakdowns of the robot in a week is at least 2 .
3. Determine the probability that the number of breakdowns of the robot in 52 weeks is less than 100.

4 At a parcel delivery company it is known that the probability that a parcel is delivered to the wrong address is 0.0005 . On a particular day, 15000 parcels are delivered. The number of parcels delivered to the wrong address is denoted by the random variable \(X\).

1. Explain why the binomial distribution and the Poisson distribution could both be suitable models for the distribution of \(X\).
2. Use a Poisson distribution to find each of the following.
   • \(\mathrm { P } ( X = 5 )\)
   • \(\mathrm { P } ( X \geqslant 8 )\)
   You are given that 15000 parcels are delivered each day in a 5-day working week.
3. 1. Determine the probability that at least 40 parcels are delivered to the wrong address during the week.
   2. Determine the probability that at least 8 parcels are delivered to the wrong address on each of the 5 days in the week.

3 A glassware factory produces a large number of ornaments each week. Just before they leave the factory, all the ornaments are checked and some may be found to be defective. The Quality Assurance Manager of the factory wishes to model the number of defective ornaments that are found each week using a Poisson distribution. The numbers of defective ornaments found each week in a period of 40 weeks are shown in Table 3.1. \begin{table}[h] \end{table} You are given that summary statistics for the data are \(\sum f = 40 , \sum \mathrm { rf } = 84\) and \(\sum \mathrm { r } ^ { 2 } \mathrm { f } = 256\).

1. By using the summary statistics to determine estimates for the mean and variance of the number of defective ornaments produced by the factory each week, explain how the data support the suggestion that the number of defective ornaments produced each week can be modelled using a Poisson distribution. The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis test for goodness of fit based on a \(\operatorname { Po } ( 2 )\) distribution.
2. Table 3.2, which is incomplete, gives observed frequency, probability, expected frequency and chi-squared contribution. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 3.2}

   No. of defective ornaments in a week, \(r\)	Observed frequency	Probability	Expected frequency	Chi-squared contribution
   0	2	0.13534	5.4134	2.15232
   1	14
   2	13	0.27067		0.43620
   3	5		7.2179
   \(\geqslant 4\)	6	0.14288		0.01421

   \end{table}
   1. Complete the copy of the table in the Printed Answer Booklet.
   2. Carry out the test at the \(10 \%\) significance level.
3. On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments and all of them are subsequently found to be defective. Explain why the Poisson model cannot model defects occurring in this manner.

1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution. \begin{table}[h] \end{table}

1. In this question you must show detailed reasoning. Use Table 1 to calculate estimates of each of the following.
   You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a).
2. Find each of the following.

4 It is known that in an electronic circuit, the number of electrons passing per nanosecond can be modelled by a Poisson distribution. In a particular electronic circuit, the mean number of electrons passing per nanosecond is 12 .

1. 1. Determine the probability that there are more than 15 electrons passing in a randomly selected nanosecond.
   2. Determine the probability that there are fewer than 50 electrons passing in a randomly selected period of 5 nanoseconds.
2. Explain what you can deduce about the electrons passing in the circuit from the fact that a Poisson distribution is a suitable model.

7 A biologist is investigating migrating butterflies. Fig. 7.1 shows the numbers of migrating butterflies passing her location in 100 randomly chosen one-minute periods. \begin{table}[h] \end{table}

1. 1. Use the data to show that a suitable estimate for the mean number of butterflies passing her location per minute is 3.3.
   2. Explain how the value of the variance estimate calculated from the sample supports the suggestion that a Poisson distribution may be a suitable model for these data. The biologist decides to carry out a test to investigate whether a Poisson distribution may be a suitable model for these data.
2. In this question you must show detailed reasoning. Complete the copy of Fig. 7.2 of expected frequencies and contributions for a chi-squared test in the Printed Answer Booklet. \begin{table}[h]

   Number of butterflies	Frequency	Probability	Expected frequency	Chi-squared contribution
   0	6	0.0369	3.6883	1.4489
   1	9	0.1217	12.1714	0.8264
   2	18			0.2160
   3	26			0.6916
   4	13	0.1823	18.2252	1.4981
   5	16	0.1203	12.0286
   6	9	0.0662	6.6158	0.8593
   \(\geqslant 7\)	3	0.0510	5.0966	0.8625

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table}
3. Complete the chi-squared test at the \(5 \%\) significance level.

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 \(100 \mathrm {~m} ^ { 2 }\).