IB DP Maths: AI HL复习笔记4.10.1 Poisson Distribution
Properties of Poisson Distribution
What is a Poisson distribution?
- A Poisson distribution is a discrete probability distribution
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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
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:
- 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
Exam Tip
- An exam question might involve different types of distributions so make it clear which distribution is being used for each variable
Worked Example
b)Calculate the standard deviation for the number of emails that Jack receives during his hour lunch breaks.
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