5.02n Sum of Poisson variables: is Poisson

6 The number of randomly occurring events in a given time interval is denoted by \(R\). In order that \(R\) is well modelled by a Poisson distribution, it is necessary that events occur independently.

1. Let \(R\) represent the number of customers dining at a restaurant on a randomly chosen weekday lunchtime. Explain what the condition 'events occur independently' means in this context, and give a reason why it would probably not hold in this context. Let \(D\) represent the number of tables booked at the restaurant on a randomly chosen day. Assume that \(D\) can be well modelled by distribution \(\operatorname { Po } ( 7 )\).
2. Find \(\mathrm { P } ( D < 5 )\).
3. Use a suitable approximation to find the probability that, in five randomly chosen days, the total number of tables booked is greater than 40 .

5 In a large region of derelict land, bricks are found scattered in the earth.

1. State two conditions needed for the number of bricks per cubic metre to be modelled by a Poisson distribution. Assume now that the number of bricks in 1 cubic metre of earth can be modelled by the distribution Po(3).
2. Find the probability that the number of bricks in 4 cubic metres of earth is between 8 and 14 inclusive.
3. Find the size of the largest volume of earth for which the probability that no bricks are found is at least 0.4.

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 $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$

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 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.

1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.

2 The number of bacteria in 1 ml of drug \(A\) has a Poisson distribution with mean 0.5. The number of the same bacteria in 1 ml of drug \(B\) has a Poisson distribution with mean 0.75 . A mixture of these drugs used to treat a particular disease consists of 1.4 ml of drug \(A\) and 1.2 ml of drug \(B\). Bacteria in the drugs will cause infection in a patient if 5 or more bacteria are injected.

1. Calculate the probability that, in a sample of 20 patients treated with the mixture, infection will occur in no more than one patient.
2. State an assumption required for the validity of the calculation.

1 The numbers of minor flaws that occur on reels of copper wire and reels of steel wire have Poisson distributions with means 0.21 per metre and 0.24 per metre respectively. One length of 5 m is cut from each reel.

1. Calculate the probability that the total number of flaws on these two lengths of wire is at least 2 .
2. State one assumption needed in the calculation.

1 On a motorway, lorries pass an observation point independently and at random times. The mean number of lorries travelling north is 6 per minute and the mean number travelling south is 8 per minute. Find the probability that at least 16 lorries pass the observation point in a given 1 -minute period.

2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

4 The number of printers, \(V\), bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, \(W\), bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4

1. Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
   [0pt] [2 marks] 4
2. State the circumstance under which the distributional model you used in part (a) would not be valid.
   [0pt] [1 mark]

3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).

1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.

2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

8

1. A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
2. The random variable \(X\) represents the number of particles emitted by a substance in a fixed time interval \(t\) minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for \(X\) if the value of \(t\) is very large.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = \mathrm { P } ( \mathrm { Y } = \mathrm { r } + 1 )\) \(\mathrm { P } ( \mathrm { Y } = \mathrm { r } ) = 1.5 \times \mathrm { P } ( \mathrm { Y } = \mathrm { r } - 1 )\). Determine the following, in either order.
   \section*{END OF QUESTION PAPER}

2 The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.

1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520.
2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval.

4 The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.

1. State a further assumption needed for the Poisson model to be valid.
2. State the value of the standard deviation of \(R\).
3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\operatorname { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
2. (a) Write down an expression for \(\mathrm { P } ( X = r )\).
   (b) Hence find \(\mathrm { P } ( X = 3 )\).
3. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.

1. (a) Given \(n\) is large, state a condition for which the binomial distribution \(\mathrm { B } ( n , p )\) can be reasonably approximated by a Poisson distribution.

A manufacturer produces candles. Those candles that pass a quality inspection are suitable for sale. It is known that 2\% of the candles produced by the manufacturer are not suitable for sale. A random sample of 125 candles produced by the manufacturer is taken.
(b) Use a suitable approximation to find the probability that no more than 6 of the candles are not suitable for sale. The manufacturer also produces candle holders.
Charlie believes that 5\% of candle holders produced by the factory have minor defects.
The manufacturer claims that the true proportion is less than \(5 \%\) To test the manufacturer's claim, a random sample of 30 candle holders is taken and none of them are found to contain minor defects.
(c) (i) Carry out a test of the manufacturer's claim using a \(5 \%\) level of significance. You should state your hypotheses clearly.
(ii) Give a reason why this is not an appropriate test. Ashley suggests changing the sample size to 50
(d) Comment on whether or not this change would make the test appropriate. Give a reason for your answer.

3. (a) State two conditions under which a Poisson distribution is a suitable model to use in statistical work. The number of cars passing an observation point in a 10 minute interval is modelled by a Poisson distribution with mean 1.
(b) Find the probability that in a randomly chosen 60 minute period there will be

1. exactly 4 cars passing the observation point,
2. at least 5 cars passing the observation point. The number of other vehicles, other than cars, passing the observation point in a 60 minute interval is modelled by a Poisson distribution with mean 12.
   (c) Find the probability that exactly 1 vehicle, of any type, passes the observation point in a 10 minute period.

1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.

The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.

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.