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

6

1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
   1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
   2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).

7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).

1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.

8

1. A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
   1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
   2. State a different condition that is needed. Explain whether it is likely to hold in this context.
2. The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
   1. In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
   2. In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
   3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$

4 In a rock, small crystal formations occur at a constant average rate of 3.2 per cubic metre.

1. State a further assumption needed to model the number of crystal formations in a fixed volume of rock by a Poisson distribution. In the remainder of the question, you should assume that a Poisson model is appropriate.
2. Calculate the probability that in one cubic metre of rock there are exactly 5 crystal formations.
3. Calculate the probability that in 0.74 cubic metres of rock there are at least 3 crystal formations.
4. Use a suitable approximation to calculate the probability that in 10 cubic metres of rock there are at least 36 crystal formations.

6 At a tourist car park, a survey is made of the regions from which cars come.

1. It is given that \(40 \%\) of cars come from the London region. Use a suitable approximation to find the probability that, in a random sample of 32 cars, more than 17 come from the London region. Justify your approximation.
2. It is given that \(1 \%\) of cars come from France. Use a suitable approximation to find the probability that, in a random sample of 90 cars, exactly 3 come from France.

4 The number of floods in a certain river plain is known to have a Poisson distribution. It is known that up until 10 years ago the mean number of floods per year was 0.32 . During the last 10 years there were 6 floods. Test at the \(1 \%\) significance level whether there is evidence of an increase in the mean number of floods per year.

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager \(A\) says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

1. Give an improved version of Manager \(A\) 's statement, and explain why the improvement is necessary.
2. Explain whether you think your improved statement is likely to hold in this context. Assume now that the number \(B\) of breakdowns per day can be modelled by the distribution \(\operatorname { Po } ( \lambda )\).
3. Given that \(\lambda = 9.0\) and \(\mathrm { P } \left( B > B _ { 0 } \right) < 0.1\), use tables to find the smallest possible value of \(B _ { 0 }\), and state the corresponding value of \(\mathrm { P } \left( B > B _ { 0 } \right)\).
4. Given that \(\mathrm { P } ( B = 2 ) = 0.0072\), show that \(\lambda\) satisfies an equation of the form \(\lambda = 0.12 \mathrm { e } ^ { k \lambda }\), for a value of \(k\) to be stated. Evaluate the expression \(0.12 \mathrm { e } ^ { k \lambda }\) for \(\lambda = 8.5\) and \(\lambda = 8.6\), giving your answers correct to 4 decimal places. What can be deduced about a possible value of \(\lambda\) ?

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .

1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
2. A page is chosen at random. Find the probability of
   (A) exactly 1 error on this page,
   (B) at least 2 errors on this page. 10 pages are chosen at random.
3. Find the probability of exactly 10 errors in these 10 pages.
4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.

2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.

1. Find the probability that John observes
   (A) no butterflies in a minute,
   (B) at least 2 butterflies in a minute,
   (C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
2. Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
3. Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
4. Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.

2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.

1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
2. Find the probability that, during one minute, Jess sees
   (A) exactly one meteor,
   (B) at least 4 meteors.
3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).

2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.

1. State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
2. State the variance of this Poisson distribution.
3. Find the probability of
   (A) exactly 3 counts in a 5 -second period,
   (B) at least 3 counts in a 5 -second period.
4. Find the probability of exactly 40 counts in a period of 60 seconds.
5. Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
6. The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.

2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.

1. Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
2. Find the probability of at most 1 car arriving in a period of 1 minute.
3. Find the probability of more than 5 cars arriving in a period of 10 minutes.
4. State the exact distribution of the number of cars arriving in a period of 1 hour.
5. Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

2 It was stated in 2012 that \(3 \%\) of \(\pounds 1\) coins were fakes. Throughout this question, you should assume that this is still the case.

1. Find the probability that, in a random selection of \(25 \pounds 1\) coins, there is exactly one fake coin. A random sample of \(250 \pounds 1\) coins is selected.
2. Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
3. Use a Poisson distribution to find the probability that, in this sample, there are
   (A) exactly 10 fake coins,
   (B) at least 10 fake coins.
4. Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins. It is known that \(0.2 \%\) of another type of coin are fakes.
5. A random sample of size \(n\) of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }\), where \(\lambda = 0.002 n\).
6. Use the approximation \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }\) for small values of \(\lambda\) to estimate the value of \(n\) for which the probability in part ( \(\mathbf { v }\) ) is equal to 0.995 .

2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
4. Use this Poisson distribution to find the probability that there are
   (A) exactly two genes that mutate,
   (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.

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.

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.