MSV 38: Adding Two Poissons

A town has two parks.

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

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

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

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).

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?

Happily, this works more generally still. If X ~ Po( ) and Y ~ Po(  ), where X and Y are independent, then X + Y ~ Po( +  ). Is it reasonable for Steve to assume that the number of accidents in each park are independent? (The spreadsheet is also on the MSV website, Activity 38.) msv/msv-38/msv-38.xlsm Adding Two Poissons spreadsheet

is written by Jonny Griffiths With thanks to pixabay.com