The Poisson distribution and its conditions Problem set A Worked solutions

A Poisson model is suggested for the number of emails that arrive at an office each hour. Write down the three conditions that must hold, in terms of the emails.

• Independent: one email arriving does not change the chance of another
• Constant average rate: the average number per hour does not change
• Singly: no two emails arrive at exactly the same moment

Texts arrive on a phone independently of each other, at a constant average rate of 5 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 texts received in the next hour.

   • X is the number of texts in a fixed hour
   • The conditions hold, so X ~ Po(5)

2. (b) X is the number of hours, out of the next 6, in which exactly 5 texts arrive.

   • X is the number of hours out of 6 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, until the next text arrives.

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

4. (d) X is the number of texts received between 10 am and 11 am tomorrow.

   • X is the number of texts in 1 hour
   • λ = 5, so X ~ Po(5)

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 arrive at a café each hour between 7 am and 11 am.

   • Condition: constant average rate
   • Customers arrive more at breakfast time than later in the morning

2. (b) The number of people in a village who catch a cold each week.

   • Condition: independence
   • If one person catches a cold, others nearby are more likely to

3. (c) The number of passengers who get off a metro train at a station each minute.

   • Condition: singly
   • A train empties many passengers at once

A lifeguard records an average of 2.4 jellyfish stings per day on a beach. X is the number of stings on the next day. Assume X is Poisson.

1. (a) Write down the distribution of X.

   • X counts stings in a fixed day, at an average rate of 2.4 per day
   • X ~ Po(2.4)

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

   • The average rate changes with the weather and the number of swimmers
   • The conditions are only roughly met, but the model is still useful

A call centre receives calls independently of each other, at a constant average rate of 3 per hour. X is the number of hours, out of the next 10, in which at least one call arrives.

1. (a) Explain why X is not modelled by a Poisson distribution.

   • X counts the intervals that meet a condition, not the calls
   • There are 10 hours, so X cannot be more than 10
   • X is not the number of events in a given period

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

   • Y counts calls in a fixed hour
   • λ = 3, so Y ~ Po(3)

Plastic particles are found in a lake at a constant average rate of 5 per litre of water, independently of each other. X is the volume of water, in litres, that must be filtered until the first particle is found.

1. (a) Explain why X is not modelled by a Poisson distribution.

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

2. (b) W is the number of particles in a 1 litre sample. Write down the distribution of W.

   • W counts particles in 1 litre
   • λ = 5, so W ~ Po(5)

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

1. (a) A fair coin is tossed 40 times. X is the number of heads.

   • Fixed number of trials: 40 tosses
   • Each is a head with probability 0.5
   • X ~ B(40, 0.5)

2. (b) Meteors cross a camera's field of view independently, at a constant average rate of 6 per hour. X is the number in the next hour.

   • Events at an average rate of 6 per hour, no fixed number of trials
   • X ~ Po(6)

3. (c) A box holds 25 bolts. Each bolt is faulty with probability 0.04, independently. X is the number of faulty bolts.

   • Fixed number of trials: 25 bolts
   • Each is faulty with probability 0.04
   • X ~ B(25, 0.04)

4. (d) Taxis arrive at a rank independently, at a constant average rate of 7 per hour. X is the number arriving between 6 pm and 7 pm.

   • Events at an average rate of 7 per hour
   • X ~ Po(7)