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Combining Poisson Distributions: Understanding Town Park Accidents

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Explore how the number of accidents in two town parks is distributed using Poisson distribution. Find the total reported park accidents in the town by adding Poisson variables X and Y. Investigate if the total accidents follow a Poisson distribution with different parameters. Access the spreadsheet for hands-on practice with Poisson distributions. Written by Jonny Griffiths from making-statistics-vital.co.uk.

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Combining Poisson Distributions: Understanding Town Park Accidents

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  1. www.making-statistics-vital.co.uk MSV 38: Adding Two Poissons

  2. A town has two parks.

  3. The number of serious accidents X in the first park over a year has been found to be distributed as Po(4).

  4. The number of serious accidents Y in the second park over a year has been found to be distributed as Po(3).

  5. Steve, a town planner, wonders, ‘How is the total number of reported park accidents in the town distributed?

  6. Let’s find P(X + Y = 2), = P(X=0, Y=2) + P(X=1, Y=1) + P(X=2, Y=0) Let’s assume that X and Y are independent... = P(X=0)P(Y=2) + P(X=1)P(Y=1) + P(X=2)P(Y=0) , where Z ~ Po(7).

  7. This leaves Steve wondering: ‘Is it the case that if X ~ Po(4) and Y ~ Po(3) and if X and Y are independent, then X + Y ~ Po(3 + 4) = Po(7)’? Can we prove this?

  8. Happily, this works more generally still. If X ~ Po(l) and Y ~ Po(m), where X and Y are independent, then X + Y ~ Po(l + m). Is it reasonable for Steve to assume that the number of accidents in each park are independent? Adding Two Poissons spreadsheet http://www.s253053503.websitehome.co.uk/msv/msv-38/msv-38.xlsm (The spreadsheet is also on the MSV website, www.making-statistics-vital.co.uk Activity 38.)

  9. With thanks to pixabay.com www.making-statistics-vital.co.uk is written by Jonny Griffiths hello@jonny-griffiths.net

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