Poisson Distribution (DP IB Maths: AI HL)

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Dan

Last updated

24 August 2023

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 $X ~ Po (m)$

m is the average rate of occurrences for the time period

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

$P (X = r) = \frac{e^{- m} m^{r}}{r!}$ for r = 0,1,2,...

e is Euler’s constant 2.718...
$r! = r \times (r - 1) \times \dots \times 2 \times 1$ and $0! = 1$
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 $X ~ Po (m)$ and $Y ~ Po (λ)$ are independent then $X + Y ~ Po (m + λ)$
- This extends to n independent Poisson distributions $X_{i} ~ Po (m_{i})$
  - $X_{1} + X_{2} + \dots + X_{n} ~ Po (m_{1} + m_{2} + \dots + m_{n})$

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: $C ~ Po (m)$

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 $Po (6.25)$ to model the number of emails he receives during his hour lunch break.

Write down two assumptions that Jack has made.

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

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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