5.02n Sum of Poisson variables: is Poisson

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]

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]

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. [3]
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. [1]

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. [2]

Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).

1. 1. Write down an expression for P\((X = r)\). [1]
   2. Hence find P\((X = 3)\). [1]
2. 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. [4]

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 \(\text{Po}(0.8)\).

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 \(\text{Po}(1.5)\). 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]