The Poisson distribution and its conditions Problem set B Worked solutions

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.

• Independent: one car passing does not change the chance of another
• Constant average rate: the average number per minute does not change
• Singly: no two cars pass at exactly the same moment

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.

   • X is the number of cyclists in a fixed hour
   • The conditions hold, so X ~ Po(12)

2. (b) X is the number of hours, out of the next 8, in which fewer than 10 cyclists pass.

   • X is the number of hours out of 8 that meet a condition
   • It counts intervals, not events, so it is not Poisson (it is binomial)

3. (c) X is the time, in minutes, between one cyclist and the next.

   • X measures the time between events
   • It is not the number of events in a given period, so it is not Poisson

4. (d) X is the number of cyclists who pass between 8 am and 9 am tomorrow.

   • X is the number of cyclists in 1 hour
   • λ = 12, so X ~ Po(12)

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.

   • Condition: constant average rate
   • Diners arrive more around 7 pm than at 10 pm

2. (b) The number of pupils in a school who catch a stomach bug each week.

   • Condition: independence
   • If one pupil catches it, others nearby are more likely to

3. (c) The number of people who enter a cinema each minute in the 20 minutes before a film starts.

   • Condition: singly
   • Groups of friends enter together, and more arrive just before the film

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.

   • X counts fallen trees in a fixed week, at an average rate of 3.1 per week
   • X ~ Po(3.1)

2. (b) Give one reason why the conditions may not be fully met.

   • A storm can bring down several trees at once, so the events are not independent
   • The conditions are only roughly met, but the model is still useful

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.

   • X counts the intervals that meet a condition, not the sign-ups
   • There are 8 hours, so X cannot be more than 8
   • X is not the number of events in a given period

2. (b) Y is the number of sign-ups in the next hour. Write down the distribution of Y.

   • Y counts sign-ups in a fixed hour
   • λ = 2, so Y ~ Po(2)

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.

   • X measures a length until an event
   • It is not the number of events in a given length, so it is not Poisson

2. (b) W is the number of flaws in a 1 metre length. Write down the distribution of W.

   • W counts flaws in 1 metre
   • λ = 4, so W ~ Po(4)

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.

   • Fixed number of trials: 30 rolls
   • Each is a six with probability 1/6
   • X ~ B(30, 1/6)

2. (b) Accidents happen at a junction independently, at a constant average rate of 1.5 per month. X is the number next month.

   • Events at an average rate of 1.5 per month, no fixed number of trials
   • X ~ Po(1.5)

3. (c) A tray holds 18 eggs. Each egg is cracked with probability 0.07, independently. X is the number of cracked eggs.

   • Fixed number of trials: 18 eggs
   • Each is cracked with probability 0.07
   • X ~ B(18, 0.07)

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.

   • Events at an average rate of 4 per trip
   • X ~ Po(4)