Poisson Distribution (DP IB Applications & Interpretation (AI)): Revision Note

A Poisson distribution is a discrete probability distribution

A discrete random variable  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  follows a Poisson distribution then it is denoted 

• m is the average rate of occurrences for the time period

The formula for the probability of r occurrences is given by:

•   for r = 0,1,2,...

  • e is Euler’s constant 2.718...

  • and 

  • 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

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  and  are independent then 

• This extends to n independent Poisson distributions 

  • 

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

Anything that satisfies the two conditions

For example, Let C be the number of calls that a helpline receives within a 15-minute period:  

• 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