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AI HL · 13.2

Probability Distributions

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13.2.10

The Poisson distribution and its conditions

13.2.10

The Poisson distribution and its conditions

Target: I can say what a Poisson variable counts and write X ~ Po(λ), where λ is the constant average rate. (+4 more)

By the end of this section

Key points

What a Poisson model is for

  • The Poisson distribution models the number of events in a fixed period of time, or a fixed length, area or volume, when the events happen at a known constant average rate.
  • If X has a Poisson distribution with constant average rate \(\lambda\), write \(X \sim \text{Po}(\lambda)\). \(\lambda\) is the average number of events in the period that X counts.
  • Three conditions: the events are independent of each other; they occur at a constant average rate; they occur singly (one at a time).
  • The formula for \(P(X = k)\) is not needed. Probabilities come from the calculator, in the next lesson.

What X must count

  • X must be the number of events in a given period. Always check this first.
  • Not Poisson: the time or the volume between events, or until the next event. X measures a size, not a count of events.
  • Not Poisson: the number of intervals (hours, pages, metres) that meet a condition. X counts intervals, and there is a fixed number of them. This is binomial.
  • The conditions can all hold and X can still fail to be Poisson. Say why: "X is not the number of events in a given period".

Poisson or binomial?

  • If you can identify a fixed number of trials, each with the same chance of success, use binomial.
  • If events happen at an average rate in time or space, with no fixed number of trials and no upper limit, use Poisson.
  • If a rate is mentioned ("on average 4 per hour"), think Poisson.

How well are the conditions met?

  • Real situations are rarely perfect. Name the condition that is least likely to hold, in the words of the context: the rate changes through the day, one event causes another, events arrive in groups.
  • A model that is not perfect can still be useful. Say so when the conditions are only roughly met.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Customers in an hour

Customers enter a shop independently of each other, at a constant average rate of 18 per hour. X is the number of customers who enter in the next hour. Is a Poisson model suitable for X? If so, write down the distribution and say how well the conditions are likely to be met.

Working
  1. What X counts

    X is the number of customers in a fixed period of 1 hour.

    A count of events in a period: a Poisson variable is possible.

  2. The conditions

    Independent: yes (stated). Constant average rate: yes (stated). Singly: customers enter one at a time, apart from groups arriving together.

    Go through the three conditions one by one.

  3. The model

    X ~ Po(18)

    λ is the average number in the period X counts: 18 per hour, for 1 hour.

  4. How well

    Likely to be met well. The rate is probably not exactly constant (busier at lunchtime), and friends often arrive together.

    Name the weak condition in the context. The model is still useful.

Video walkthrough coming
Time between orders

An online shop receives orders independently of each other, at a constant average rate of 9 per hour. X is the time, in minutes, between one order and the next. Is a Poisson model suitable for X?

Working
  1. The conditions

    Independent: yes. Constant average rate: yes. Singly: yes.

    All three conditions are met.

  2. What X measures

    X is a time, not a number of orders.

    A Poisson variable counts events. It never measures a time.

  3. Conclusion

    No: X is not the number of events in a given time period.

    The number of orders in a given hour would be Po(9). The time between orders is a different distribution.

Video walkthrough coming
Faultless lengths of cloth

A loom makes faults in cloth independently of each other, at a constant average rate of 1.5 faults per metre. A quality inspector examines 25 separate one-metre lengths. X is the number of these lengths that have no faults. Is a Poisson model suitable for X?

Working
  1. The conditions

    The faults are independent, at a constant average rate, and occur singly.

    The faults themselves could be modelled by a Poisson distribution.

  2. What X counts

    X counts lengths of cloth (intervals) that meet a condition, not faults.

    There are 25 lengths, each with or without faults.

  3. Conclusion

    No: X is not the number of events in a given period. X has a fixed number of trials (25 lengths), so it is binomial.

    A Poisson variable has no upper limit. This X cannot be more than 25.

Video walkthrough coming
Patients in an emergency department

An emergency department admits patients at an average rate of 5.5 per hour. X is the number of patients admitted between 2 pm and 3 pm on a weekday. Is a Poisson model suitable for X? Comment on the conditions.

Working
  1. What X counts

    X is the number of admissions in a fixed hour.

    A count of events in a period.

  2. Conditions

    Independent: mostly, but one accident can bring several patients. Constant rate: roughly, for a single hour. Singly: mostly.

    Choose the condition that is least likely to hold, and say why.

  3. Conclusion

    X ~ Po(5.5) is a reasonable model, though the conditions are only roughly met.

    A short, quiet hour fits better than a whole day, because the rate changes less.

Video walkthrough coming