5.02m Poisson: mean = variance = lambda

5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)

1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
2. Find the largest possible value of \(k\) In a randomly selected month,
3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
4. Find the probability that at least 10 packages were lost in each of these two months.
5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

1. (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company's order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\) Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of \(\alpha\) (d) the correct probability.

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
3. 1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
   2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
   

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1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528

1. Find the value of \(x\)
2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.

The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75

1. Find the largest possible value of \(v\)
2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
   

2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)

1. 1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
   2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)

4. In a large population, past records show that 1 in 200 adults has a particular allergy. In a random sample of 700 adults selected from the population, estimate

1. 1. the mean number of adults with the allergy,
   2. the standard deviation of the number of adults with the allergy. Give your answer to 3 decimal places. A doctor claims that the past records are out of date and the proportion of adults with the allergy is higher than the records indicate. A random sample of 500 adults is taken from the population and 5 are found to have the allergy. A test of the doctor's claim is to be carried out at the 5\% level of significance.
2. 1. State the hypotheses for this test.
   2. Using a suitable approximation, carry out the test.
      (6) It is also claimed that \(30 \%\) of those with the allergy take medication for it daily. To test this claim, a random sample of \(n\) people with the allergy is taken. The random variable \(Y\) represents the number of people in the sample who take medication for the allergy daily. A two-tailed test, at the \(1 \%\) level of significance, is carried out to see if the proportion differs from 30\% The critical region for the test is \(Y = 0\) or \(Y \geqslant w\)
3. Find the smallest possible value of \(n\) and the corresponding value of \(w\)

2. John weaves cloth by hand. Emma believes that faults are randomly distributed in John's cloth at a rate of more than 4 per 50 metres of cloth. To check her belief, Emma takes a random sample of 100 metres of the cloth and finds that it contains 14 faults.

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, Emma's belief. Armani also weaves cloth by hand. He knows that faults are randomly distributed in his cloth at a rate of 4 per 50 metres of cloth. Emma decides to buy a large amount of Armani's cloth to sell in pieces of length \(l\) metres. She chooses \(l\) so that the probability of no faults in a piece is exactly 0.9
2. Show that \(l = 1.3\) to 2 significant figures. Emma sells 5000 of these pieces of cloth of length 1.3 metres. She makes a profit of \(\pounds 2.50\) on each piece of cloth that does not contain any faults but a loss of \(\pounds 0.50\) on any piece that contains at least one fault.
3. Find Emma's expected profit.

5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.

1. Calculate the probability that in a randomly selected 10 minute period,
   1. exactly 7 cars will stop at the service station,
   2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
2. Find the value of \(n\).

1. The independent random variables \(W\) and \(X\) have the following distributions.

$$W \sim \operatorname { Po } ( 4 ) \quad X \sim \mathrm {~B} ( 3,0.8 )$$

1. Write down the value of the variance of \(W\)
2. Determine the mode of \(X\) Show your working clearly. One observation from each distribution is recorded as \(W _ { 1 }\) and \(X _ { 1 }\) respectively.
3. Find \(\mathrm { P } \left( W _ { 1 } = 2 \right.\) and \(\left. X _ { 1 } = 2 \right)\)
4. Find \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)

3. The random variable \(X\) is the number of misprints per page in the first draft of a novel.

1. State two conditions under which a Poisson distribution is a suitable model for \(X\). The number of misprints per page has a Poisson distribution with mean 2.5. Find the probability that
2. a randomly chosen page has no misprints,
3. the total number of misprints on 2 randomly chosen pages is more than 7 . The first chapter contains 20 pages.
4. Using a suitable approximation find, to 2 decimal places, the probability that the chapter will contain less than 40 misprints.

4

1. A discrete random variable \(X\) has a probability function defined by $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } \quad \text { for } x = 0,1,2,3,4 , \ldots \ldots$$
   1. State the name of the distribution of \(X\).
   2. Write down, in terms of \(\lambda\), expressions for \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
   3. Write down an expression for \(\mathrm { P } ( X = x + 1 )\), and hence show that $$\mathrm { P } ( X = x + 1 ) = \frac { \lambda } { x + 1 } \mathrm { P } ( X = x )$$
2. The number of cars and the number of coaches passing a certain road junction may be modelled by independent Poisson distributions.
   1. On a winter morning, an average of 500 cars per hour and an average of 10 coaches per hour pass this junction. Determine the probability that a total of at least 10 such vehicles pass this junction during a particular 1 -minute interval on a winter morning.
   2. On a summer morning, an average of 836 cars per hour and an average of 22 coaches per hour pass this junction. Calculate the probability that a total of at most 3 such vehicles pass this junction during a particular 1 -minute interval on a summer morning. Give your answer to two significant figures.
      (3 marks)

4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.

1. Find the probability that the satellite detects:
   1. exactly 4 GRBs during one particular week;
   2. at least 2 GRBs on one particular day;
   3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
2. Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
   (1 mark)

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]

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]

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.

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 \(\lambda = 2\)

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.

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.