Poisson distribution

It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8. Find the critical region to test the hypothesis \(H_0 : \lambda = 8\) against the hypothesis \(H_1 : \lambda < 8\), working at the 1\% significance level. [5 marks]

4. In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)

1. Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per \(\mathrm { m } ^ { 2 }\) of the peat bog to follow a Poisson distribution.
   (1) Given that the number of Common Spotted-orchids in \(1 \mathrm {~m} ^ { 2 }\) of the peat bog can be modelled by a Poisson distribution,
2. find the probability that in a randomly selected \(1 \mathrm {~m} ^ { 2 }\) of the peat bog
   1. there are exactly 6 Common Spotted-orchids,
   2. there are fewer than 10 but more than 4 Common Spotted-orchids.
      (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected \(2 \mathrm {~m} ^ { 2 }\) of the peat bog contains 11 Common Spotted-orchids.
3. Using a \(5 \%\) significance level assess Juan's belief. State your hypotheses clearly. Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
4. use a normal approximation to find the probability that in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog there are fewer than 70 Common Spotted-orchids. Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog is 0.012 A random sample of 200 non-overlapping \(20 \mathrm {~m} ^ { 2 }\) areas of the peat bog is taken.
5. Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-15_2255_50_314_34}

5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8 Emily conducts a hypothesis test to investigate her claim.
5

1. State the hypotheses for Emily's test. 5
2. Emily counts the number of runners, \(X\), passing the shop in a randomly chosen minute. The critical region for Emily's test is \(X \leq 2\) or \(X \geq 14\) During a randomly chosen minute, Emily counts 3 runners passing the shop.
   Determine the outcome of Emily's hypothesis test.
   5
3. The actual average number of runners per minute passing the shop is 7 Find the power of Emily's hypothesis test, giving your answer to three significant figures.

6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be

1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
2. fewer than 1000 calls in four weeks ( 672 hours).

3. A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. Assuming this model and a mean of 0.7 weeds per \(\mathrm { m } ^ { 2 }\), find
2. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
3. Using a suitable approximation, find the probability that in a plot of \(100 \mathrm {~m} ^ { 2 }\) there will be more than 66 of these weeds.
   (6)

6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that

1. at least 4 customers arrive in the next 10 minutes,
2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END

Naomi produces oak tabletops, each of area 4·8 m². Defects in the oak tabletops occur randomly at a rate of 0·25 per m².

1. Find the probability that a randomly chosen tabletop will contain at most 2 defects. [3]
2. Find the probability that, in a random sample of 7 tabletops, exactly 4 will contain at most 2 defects each. [3]

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 \(\mathrm { P } \left( X _ { 1 } < W _ { 1 } \right)\)

1. On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.
   1. Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
   2. Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday.
   The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
   She records the number of calls received in each of these 30-minute periods.
2. Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
3. Explain why the Poisson approximation may be reasonable in this case. The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
4. Write down the hypotheses for this test. The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
5. Carry out the test using a \(5 \%\) level of significance.

1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.

1. Use a Poisson distribution to find the probability that, at a given moment,
   1. in a randomly chosen area of 3 acres there are at least 4 foxes,
   2. in a randomly chosen area of 1 acre there are exactly 2 foxes.
   3. Explain briefly why a Poisson distribution might not be a suitable model.

The small village of Tornep has a preservation society which is campaigning for a new by-pass to be built. The society needs to measure

1. the strength of opinion amongst the residents of Tornep for the scheme and
2. the flow of traffic through the village on weekdays. The society wants to know whether to use a census or a sample survey for each of these measures.
   1. In each case suggest which they should use and specify a suitable sampling frame. [4] For the measurement of traffic flow through Tornep,
   2. suggest a suitable statistic and a possible statistical model for this statistic. [2]

3 A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.

1. Assume that the owner is correct.
   1. Find the probability that there will be at least 4 hits during a 10-minute period.
   2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.
      A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
2. 1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
   2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.

6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]

It is given that \(Y \sim\) Po\((\lambda)\), where \(\lambda \neq 0\), and that P\((Y = 4) =\) P\((Y = 5)\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of P\((Y = 5)\). [5]

6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.

1. For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
2. For another path the mean of \(X\) is 1.3.
   1. On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
   2. Find the probability that there are more than 1250 worms on a one kilometre length of this path.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).

1 The random variable \(X\) has a Poisson distribution with mean 16 Find the standard deviation of \(X\) Circle your answer.
4
8
16
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8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.

1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.

4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.

1. Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
2. Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.

4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).

1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.

4

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).

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

1. Find the probability that at least 28 orders are received in a randomly chosen 10-minute period. [2]
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]
3. State a necessary assumption for the validity of your calculation in part (b). [1]

5 Over a long period of time, it is found that the mean number of mistakes made by a certain player when playing a particular piece of music is 5 . The number of mistakes that the player makes when playing the piece is denoted by the random variable \(Y\).

1. State two assumptions necessary for \(Y\) to be modelled by a Poisson distribution. For the remainder of this question you may assume that \(Y\) can be modelled by a Poisson distribution.
2. 1. Find the probability that the player makes exactly 3 mistakes when playing the piece.
   2. Find the probability that the player makes fewer than 3 mistakes when playing the piece.
   3. Find the probability that the player makes fewer than 6 mistakes in total when playing the piece twice, assuming that the performances are independent. In a recording studio, the player plays the piece once in the morning and once in the afternoon each day for one week (7 days). It can be assumed that all the performances are independent of each other. The performances are recorded onto two CDs, one for each of two critics, A and B, to review. The critics are interested in the total number of mistakes made by the player per day. Unfortunately, there is a recording error in one of the CDs; on this CD, every piece that is supposed to be an afternoon recording is in fact just a repeat of that morning's recording. The random variables \(M _ { 1 }\) and \(M _ { 2 }\) represent the total number of mistakes per day for the correctly recorded CD and for the wrongly recorded CD respectively.
3. By considering the values of \(\mathrm { E } \left( M _ { 1 } \right)\) and \(\mathrm { E } \left( M _ { 2 } \right)\) explain why it is not possible to use the mean number of mistakes per day on the CDs to determine which critic received the wrongly recorded CD. Each critic counts the total number of mistakes made per day, for each of the 7 days of recordings on their CD. Summary data for this is given below. Critic A: \(\quad n = 7 , \quad \sum x _ { A } = 70 , \quad \sum x _ { A } ^ { 2 } = 812\) Critic B: \(\quad \mathrm { n } = 7 , \sum \mathrm { x } _ { \mathrm { B } } = 72 , \sum \mathrm { x } _ { \mathrm { B } } ^ { 2 } = 800\)
4. By considering the values of \(\operatorname { Var } \left( M _ { 1 } \right)\) and \(\operatorname { Var } \left( M _ { 2 } \right)\) determine which critic is likely to have received the wrongly recorded CD.

1. A bakery sells muffins individually at an average rate of 8 muffins per hour.
   1. Find the probability that, in a randomly selected one-hour period, the bakery sells at least 4 but not more than 8 muffins.
   A sample of 5 non-overlapping half-hour periods is selected at random.
2. Find the probability that the bakery sells fewer than 3 muffins in exactly 2 of these periods. Given that 4 muffins were sold in a one-hour period,
3. find the probability that more muffins were sold in the first 15 minutes than in the last 45 minutes.

In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km\(^2\).

1. Find the probability that in a randomly selected area of 1000 km\(^2\) there are at least 10 barn owls. [2]
2. Find the probability that in a randomly selected area of 200 km\(^2\) there are exactly 2 barn owls. [3]
3. Using a suitable approximation, find the probability that in a randomly selected area of 50000 km\(^2\) there are at least 470 barn owls. [6]

An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]

Find the probability that in a randomly chosen hour

1. all Internet users connect at their first attempt, [2]
2. more than 4 users fail to connect at their first attempt. [2]

1. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
2. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [6]

6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of

1. exactly 4 calls in an 8 -minute period,
2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.

The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the 5% significance level. [10]

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.

2 Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws needed is recorded and the results are summarised in the following table. Carry out a goodness of fit test, at the \(5\%\) significance level, to test whether \(\operatorname{Geo}(0.6)\) is a satisfactory model for the data.

1 A roller-coaster ride has a safety system to detect faults on the track.

1. State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
2. Find the probability that on a randomly chosen day there are
   (A) no faults,
   (B) at least 2 faults.
3. Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
4. State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
   [0pt]
5. The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]

6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.

1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
   (b) Justify the approximating distribution used in part (ii)(a).
3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .

4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.

1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .

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.

4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.

1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .

1. 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.

6 A statistician investigates the number, \(F\), of signal failures per week on a railway network.

1. The statistician assumes that signal failures occur randomly. Explain what this statement means.
2. State two further assumptions needed for \(F\) to be well modelled by a Poisson distribution. In a random sample of 50 weeks, the statistician finds that the mean number of failures per week is 1.61, with standard deviation 1.28.
3. Explain whether this suggests that \(F\) is likely to be well modelled by a Poisson distribution. Assume first that \(F \sim \operatorname { Po } ( 1.61 )\).
4. Write down an exact expression for \(\mathrm { P } ( F = 0 )\).
5. Complete the table in the Printed Answer Booklet to show the probabilities of different values of \(F\), correct to three significant figures.

   Value of \(F\)	0	1	\(\geqslant 2\)
   Probability	0.200

   After further investigation, the statistician decides to use a different model for the distribution of \(F\). In this model it is now assumed that \(\mathrm { P } ( F = 0 )\) is still 0.200 , but that if one failure occurs, there is an increased probability that further failures occur.
6. Explain the effect of this assumption on the value of \(\mathrm { P } ( F = 1 )\).

8 A company records the number of complaints, \(X\), that it receives over 60 months. The summarised results are $$\sum x = 102 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 103.25$$ 8

1. Using this data, explain why it may be appropriate to model the number of complaints received by the company per month by a Poisson distribution with mean 1.7
   8
2. The company also receives enquiries as well as complaints. The number of enquiries received is independent of the number of complaints received. The company models the number of complaints per month with a Poisson distribution with mean 1.7 and the number of enquiries per month with a Poisson distribution with mean 5.2 The company starts selling a new product.
   The company records a total of 3 complaints and enquiries in one randomly chosen month. Investigate if the mean total number of complaints and enquiries received per month has changed following the introduction of the new product, using the \(10 \%\) level of significance.
   8
3. It is later found that the mean total number of complaints and enquiries received per month is 6.1 Find the power of the test carried out in part (b), giving your answer to four decimal places. \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-15_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-20_2496_1723_214_148}

Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\). Find the probability that the total number of accidents in the two factories during a 2-month period is more than 3. [4]

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]

Another machine produces biscuits with a mean of 1.9 currants per biscuit.

1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]

3 In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.

1. Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.
2. Justify the use of your approximating distribution in this context.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 1.3 )\). The random variable \(Y\) is defined by \(Y = 2 X\).

1. Find the mean and variance of \(Y\).
2. Give a reason why the variable \(Y\) does not have a Poisson distribution.

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

1. Prove that \(\mathrm { E } ( X ) = \lambda\).
2. By first proving that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), or otherwise, prove that \(\operatorname { Var } ( X ) = \lambda\).

On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407, to 3 significant figures. [1] Find the probability of
2. exactly 2 accidents in the next 6 month period, [3]
3. no accidents in exactly 2 of the next 4 months. [3]

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.

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.

6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.

1. Find the probability that exactly two people arrive in a 1-minute period.
2. Find the probability that nobody arrives in a 15 -second period.
3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).

\(X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.

1. Find P\((X + Y > 3)\). [4]
2. A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4. [4]

1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
   1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
      1. exactly 7
      2. at least 10
   A patient arrives at 11.30 a.m.
2. Find the probability that the next patient arrives before 11.45 a.m.

The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.

1. Find the probability exactly 4 calls in a 10 minute period. [2]
2. Find the probability at least 3 calls in a 3 minute period. [3]

4 On average, 1 in 2500 people have a particular gene.

1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).

1 On average, 1 in 50000 people have a certain gene.
Use a suitable approximating distribution to find the probability that more than 2 people in a random sample of 150000 have the gene.

1. The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120
   1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box.
   A retailer buys 2 boxes of components.
2. Estimate the probability that there are at least 4 defective components in each box.

1. In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9
   1. Find
      1. \(\mathrm { P } ( X > 5 )\)
      2. \(\mathrm { P } ( 4 \leqslant X < 10 )\)
   The length of a working day is 7 hours.
2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

2 A special railway coach detects faults in the railway track before they become dangerous.

1. Write down the conditions required for the numbers of faults in the track to be modelled by a Poisson distribution. You should now assume that these conditions do apply, and that the mean number of faults in a 5 km length of track is 1.6 .
2. Find the probability that there are at least 2 faults in a randomly chosen 5 km length of track.
3. Find the probability that there are at most 10 faults in a randomly chosen 25 km length of track.
4. On a particular day the coach is used to check 10 randomly chosen 1 km lengths of track. Find the probability that exactly 1 fault, in total, is found.

A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,

1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page. [1 mark]
2. Find the probability that a particular page has more than 2 misprints. [3 marks]
3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]

Chapter 2 is longer, at 20 pages.

1. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. [7 marks]

5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).

1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
   Use your approximating distribution to answer parts (b) and (c).
2. Find \(\mathrm { P } ( X \leqslant 3 )\).
3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
   The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).

6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6

1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
   6
3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
   6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.

6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be

1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
2. fewer than 1000 calls in four weeks ( 672 hours).

4 Goals scored by Femchester United occur at random with a constant average of 1.2 goals per match. Goals scored against Femchester United occur independently and at random with a constant average of 0.9 goals per match.

1. Find the probability that in a randomly chosen match involving Femchester,
   1. a total of 3 goals are scored,
   2. a total of 3 goals are scored and Femchester wins. The manager promises the Femchester players a bonus if they score at least 35 goals in the next 25 matches.
   3. Find the probability that the players receive the bonus.