5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

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]

2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.

1. Find the probability that he sells no copies in a particular week.
2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.

7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,

1. find the probability that the chosen patch contains
   1. no daisies,
   2. one daisy. Ten such patches are chosen. Using your answers to part (a),
2. find the probability that the total number of daisies is less than two.
3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
4. Compare your answers to parts (b) and (c).
5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.

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}

1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
   (3 marks)

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)

7

1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
   1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
   2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
      [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}

2 Emilia runs an online perfume business from home. She believes that she receives more orders on Mondays than on Fridays. She checked this during a period of 26 weeks and found that she received a total of 507 orders on the Mondays and a total of 416 orders on the Fridays. The daily numbers of orders that Emilia receives may be modelled by independent Poisson distributions with means \(\lambda _ { \mathrm { M } }\) for Mondays and \(\lambda _ { \mathrm { F } }\) for Fridays.

1. Construct an approximate \(99 \%\) confidence interval for \(\lambda _ { \mathrm { M } } - \lambda _ { \mathrm { F } }\).
2. Hence comment on Emilia's belief.

5

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find an expression for \(\operatorname { Var } ( X )\).
2. 1. The random variable \(Y\) has a binomial distribution with \(\mathrm { E } ( Y ) = 3\) and \(\operatorname { Var } ( Y ) = 2.985\). Find values for \(n\) and \(p\).
   2. The random variable \(U\) has \(\mathrm { E } ( U ) = 5\) and \(\operatorname { Var } ( U ) = 6.25\). Show that \(U\) does not have a binomial distribution.
3. The random variable \(V\) has the distribution \(\operatorname { Po } ( 5 )\) and \(W = 2 V + 10\). Show that \(\mathrm { E } ( W ) = \operatorname { Var } ( W )\) but that \(W\) does not have a Poisson distribution.
4. The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
   [0pt] [3 marks]

3. It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 8\) against \(\mathrm { H } _ { 1 } : \lambda \neq 8\), using a \(10 \%\) level of significance.

1. Find the critical region for this test, such that the probability in each tail is as close as possible to \(5 \%\).
2. Given that \(\lambda = 10\), find
   1. the probability of a type II error,
   2. the power of the test.
      (4)

5. Rolls of cloth delivered to a factory contain defects at an average rate of \(\lambda\) per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that \(\lambda > 0.3\). The criterion that the manager uses for rejecting the hypothesis that \(\lambda = 0.3\) is that there are 9 or more defects in the sample.

1. Find the size of the test. Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.15	0.34	\(r\)	0.72	0.85	0.92	0.96

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find the value of \(r\). The manager would like to design a test, of whether or not \(\lambda > 0.3\), that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than \(10 \%\).
3. Find the criterion to reject the hypothesis that \(\lambda = 0.3\) which makes the test as powerful as possible.
4. Hence state the size of this second test. Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c). \begin{table}[h]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.21	0.38	0.55	0.70	\(s\)	0.88	0.93

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
5. Find the value of \(s\).
6. Using the same axes, on graph paper draw the graphs of the power functions of these two tests.
7. 1. State the value of \(\lambda\) where the graphs cross.
   2. Explain the significance of \(\lambda\) being greater than this value. The cost of wrongly rejecting a delivery of cloth with \(\lambda = 0.3\) is low. Deliveries of cloth with \(\lambda > 0.7\) are unusual.
8. Suggest, giving your reasons, which the test manager should adopt.
   (2)

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 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.