The Poisson distribution and its conditions Problem set A

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.

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.
2. (b) X is the number of hours, out of the next 6, in which exactly 5 texts arrive.
3. (c) X is the time, in minutes, until the next text arrives.
4. (d) X is the number of texts received between 10 am and 11 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 arrive at a café each hour between 7 am and 11 am.
2. (b) The number of people in a village who catch a cold each week.
3. (c) The number of passengers who get off a metro train at a station each minute.

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

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

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.
2. (b) W is the number of particles in a 1 litre sample. 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 coin is tossed 40 times. X is the number of heads.
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.
3. (c) A box holds 25 bolts. Each bolt is faulty with probability 0.04, independently. X is the number of faulty bolts.
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.