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IBMaths: AI HLDPRevision Notes4. Statistics & Probability4.10 Poisson Distribution4.10.1 Poisson Distribution

Poisson Distribution (DP IB Maths: AI HL)

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Dan

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Dan

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Properties of Poisson Distribution

What is a Poisson distribution?

  • A Poisson distribution is a discrete probability distribution
  • A discrete random variable X follows a Poisson distribution if it counts the number of occurrences in a fixed time period given the following conditions:
    • Occurrences are independent
    • Occurrences occur at a uniform average rate for the time period (m)
  • If X follows a Poisson distribution then it is denoted X blank tilde Po invisible function application open parentheses m close parentheses
    • m is the average rate of occurrences for the time period
  • The formula for the probability of r occurrences is given by:
    •  straight P invisible function application open parentheses X equals r close parentheses equals fraction numerator straight e to the power of negative m end exponent m to the power of r over denominator r factorial end fraction for r = 0,1,2,...
      • e is Euler’s constant 2.718...
      • r factorial equals r cross times open parentheses r minus 1 close parentheses cross times horizontal ellipsis cross times 2 cross times 1 and 0 factorial equals 1
      • There is no upper bound for the number of occurrences
    • You will be expected to use the distribution function on your GDC to calculate probabilities with the Poisson distribution

What are the important properties of a Poisson distribution?

  • The expected number (mean) of occurrences is m
    • You are given this in the formula booklet
  • The variance of the number of occurrences is m
    • You are given this in the formula booklet
    • Square root to get the standard deviation
  • The mean and variance for a Poisson distribution are equal
  • The distribution can be represented visually using a vertical line graph
    • The graphs have tails to the right for all values of m
    • As m gets larger the graph gets more symmetrical
  • If X tilde Po invisible function application open parentheses m close parentheses and Y tilde Po invisible function application open parentheses lambda close parentheses are independent then X plus Y tilde Po invisible function application open parentheses m plus lambda close parentheses
    • This extends to n independent Poisson distributions X subscript i blank tilde Po invisible function application open parentheses m subscript i close parentheses
      • X subscript 1 plus X subscript 2 plus blank horizontal ellipsis plus X subscript n blank tilde Po invisible function application open parentheses m subscript 1 plus m subscript 2 plus blank horizontal ellipsis plus m subscript n close parentheses

2-1-1-poisson-distribution-diagram-1

Modelling with Poisson Distribution

How do I set up a Poisson model?

  • Identify what an occurrence is in the scenario
    • For example: a car passing a camera, a machine producing a faulty item
  • Use proportion to find the mean number of occurrences for the given time period
    • For example: 10 cars in 5 minutes would be 120 cars in an hour if the Poisson model works for both time periods
  • Make sure you clearly state what your random variable is
    • For example: let X be the number of cars passing a camera in 10 minutes

What can be modelled using a Poisson distribution?

  • Anything that satisfies the two conditions
  • For example, Let C be the number of calls that a helpline receives within a 15-minute period:  C tilde Po left parenthesis m right parenthesis
    • An occurrence is the helpline receiving a call and can be considered independent
    • The helpline receives calls at an average rate of m calls during a 15-minute period
  • Sometimes a measure of space will be used instead of a time period
    • For example, the number of daisies that exist on a square metre of grass
  • If the mean and variance of a discrete variable are equal then it might be possible to use a Poisson model

Examiner Tip

  • An exam question might involve different types of distributions so make it clear which distribution is being used for each variable

Worked example

Jack uses Po left parenthesis 6.25 right parenthesis  to model the number of emails he receives during his hour lunch break.

a)
Write down two assumptions that Jack has made.

4-10-1-ib-ai-hl-poisson-model-a-we-solution

b)
Calculate the standard deviation for the number of emails that Jack receives during his hour lunch breaks.

4-10-1-ib-ai-hl-poisson-model-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.