MSV 38: Adding Two Poissons

www.making-statistics-vital.co.uk

A town has two parks.

The number of serious accidents X in the first parkover 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, www.making-statistics-vital.co.uk Activity 38.)

http://www.s253053503.websitehome.co.uk/msv/

msv-38/msv-38.xlsm

Adding Two Poissonsspreadsheet

www.making-statistics-vital.co.uk

is written by Jonny Griffiths

With thanks to pixabay.com

hello@jonny-griffiths.net