The Poisson distribution and its conditions Problem set B

A Poisson model is suggested for the number of cars that pass a point on a quiet road each minute. Write down the three conditions that must hold, in terms of the cars.

Cyclists pass a checkpoint independently of each other, at a constant average rate of 12 per hour. Say whether X can be modelled by a Poisson distribution. If it can, write X ~ Po(λ). If it cannot, say why not.

1. (a) X is the number of cyclists who pass in the next hour.
2. (b) X is the number of hours, out of the next 8, in which fewer than 10 cyclists pass.
3. (c) X is the time, in minutes, between one cyclist and the next.
4. (d) X is the number of cyclists who pass between 8 am and 9 am tomorrow.

Each of these is to be modelled by a Poisson distribution. Say which of the three conditions is least likely to hold, and why.

1. (a) The number of customers who enter a restaurant each hour between 5 pm and 10 pm.
2. (b) The number of pupils in a school who catch a stomach bug each week.
3. (c) The number of people who enter a cinema each minute in the 20 minutes before a film starts.

A park ranger records an average of 3.1 fallen trees per week along a woodland trail. X is the number of fallen trees next week. Assume X is Poisson.

1. (a) Write down the distribution of X.
2. (b) Give one reason why the conditions may not be fully met.

A website receives sign-ups independently of each other, at a constant average rate of 2 per hour. X is the number of hours, out of the next 8, in which no sign-ups arrive.

1. (a) Explain why X is not modelled by a Poisson distribution.
2. (b) Y is the number of sign-ups in the next hour. Write down the distribution of Y.

A machine makes flaws in a long strip of metal at a constant average rate of 4 per metre, independently of each other. X is the length of strip, in metres, that must be inspected until the first flaw is found.

1. (a) Explain why X is not modelled by a Poisson distribution.
2. (b) W is the number of flaws in a 1 metre length. Write down the distribution of W.

For each situation, decide whether X is binomial or Poisson, and write down the distribution.

1. (a) A fair die is rolled 30 times. X is the number of sixes.
2. (b) Accidents happen at a junction independently, at a constant average rate of 1.5 per month. X is the number next month.
3. (c) A tray holds 18 eggs. Each egg is cracked with probability 0.07, independently. X is the number of cracked eggs.
4. (d) Whales are seen from a ferry independently, at a constant average rate of 4 per trip. X is the number seen on the next trip.