5.02i Poisson distribution: random events model

The random variable \(R\) has the distribution Po\((\lambda)\). A significance test is carried out at the 1% level of the null hypothesis H\(_0\): \(\lambda = 11\) against H\(_1\): \(\lambda > 11\), based on a single observation of \(R\). Given that in fact the value of \(\lambda\) is 14, find the probability that the result of the test is incorrect, and give the technical name for such an incorrect outcome. You should show the values of any relevant probabilities. [6]

An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.

1. Show that the sample mean is 0.4375. [1]
2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]

The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.

1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]

The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.

1. Assuming that free repairs are required independently, find the probability that
   1. the two appliances need a total of 3 free repairs between them,
   2. each appliance needs exactly one free repair.
   [5]

An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players. [3 marks]
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]

A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.

1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed. [1 mark]
2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
3. Find the probability that on a weekday morning the shop sells
   1. more than 4 pairs in a one-hour period,
   2. no pairs in a half-hour period,
   3. more than 4 pairs during each hour from 9 am until noon. [6 marks]

The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop's sales.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of an increase in sales following the appearance of the advert. [4 marks]

An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.

1. Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
2. Explain why the parameter used with this model may need to be changed if
   1. 50 g of nuts are used instead of breadcrumbs,
   2. 100g of breadcrumbs are used.
   [2 marks]

A bird table in a rural garden has 50 g of breadcrumbs spread on it. Find the probability that

1. exactly 6 different species visit the table, [2 marks]
2. more than 2 different species visit the table. [4 marks]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives

1. 3 complaints, [3 marks]
2. 10 or more complaints. [2 marks]

The supermarket is open on six days each week.

1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]

A rugby player scores an average of 0.4 tries per match in which he plays.

1. Find the probability that he scores 2 or more tries in a match. [5 marks]

The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]

A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008

1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]

A charity receives donations of more than £10000 at an average rate of 25 per year. Find the probability that the charity receives

1. exactly 30 such donations in one year, [3]
2. less than 3 such donations in one month. [5]
3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than £10000 in the next two years. [5]

A student collects data on the number of bicycles passing outside his house in 5-minute intervals during one morning.

1. Suggest, with reasons, a suitable distribution for modelling this situation. [3]

The student's data is shown in the table below.

1. Show that the mean and variance of these data are 1.5 and 1.58 (to 3 significant figures) respectively and explain how these values support your answer to part (a). [6]

An environmental organisation declares a "car free day" encouraging the public to leave their cars at home. The student wishes to test whether or not there are more bicycles passing along his road on this day and records 16 bicycles in a half-hour period during the morning.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there are more than 1.5 bicycles passing along his road per 5-minute interval that morning. [5]

1. Define
   1. a Type I error,
   2. a Type II error. [2]

A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.

1. [(b)] Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
2. Calculate the probability of the Type I error for this test. [3]
3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]

It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses H₀: \(\lambda = 8\) against H₁: \(\lambda \neq 8\), using a 10\% level of significance.

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

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

1. State these two assumptions. [2]
2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\mathrm{Po}(0.8)\).

1. 1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
   2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\mathrm{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]

The numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\)? [2]

Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).

1. Find
   1. P\((X = 6)\), [2]
   2. P\((X \geqslant 8)\). [2]

The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).

1. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
2. State an assumption needed for your answer to part (c) to be valid. [1]
3. Give a reason why the assumption in part (d) may not be valid in practice. [1]

1. State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day. [2]

Jane records the number of junk emails which she receives each day. During working hours (\(9\)am to \(5\)pm, Monday to Friday) the mean number of junk emails is \(7.4\) per day. Outside working hours (\(5\)pm to \(9\)am), the mean number of junk emails is \(0.3\) per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".

1. Find the probability that the number of junk emails which she receives between \(9\)am and \(5\)pm on a Monday is
   1. exactly \(10\), [1]
   2. at least \(10\). [2]
2. 1. What assumption must you make to calculate the probability that the number of junk emails which she receives from \(9\)am Monday to \(9\)am Tuesday is at most \(20\)? [1]
   2. Find the probability. [2]

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]

A baker sells 3-5 birthday cakes per hour on average.

1. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution. [1]
2. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period. [2]
3. Calculate the probability that, during a randomly selected 3-hour period, the baker sells more than 10 birthday cakes. [3]
4. The baker sells a birthday cake at 9:30 a.m. Calculate the probability that the baker will sell the next birthday cake before 10:00 a.m. [3]
5. Select one of the assumptions in part (a) and comment on its reasonableness. [1]

Cars arrive at random at a toll bridge at a mean rate of 15 per hour.

1. Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval. [1]
2. Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive. [3]
3. Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3. [3]

Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).

1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
2. A customer arrives at exactly 11 a.m.
   1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
   2. Hence find the probability density function, \(f(t)\), of \(T\).
   3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]

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]

A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.

1. Find the probability of exactly 4 faults in a 15 metre length of cloth. (2)
2. Find the probability of more than 10 faults in 60 metres of cloth. (3)

A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80

1. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. (4)

The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.

1. Find the retailer's expected profit. (4)

Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.

1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20-minute period one morning. [3]

Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5-minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.

1. Find the probability that in exactly 3 of these periods there were no calls. [2]

On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.

1. Find the probability that Indre missed exactly 1 call in each of these 2 breaks. [3]