The Poisson Distribution (Edexcel IAL Maths)

Properties of Poisson Distribution

Spec point — 1.1

What is a Poisson distribution?

• A Poisson distribution is a discrete probability distribution

• The discrete random variable X follows a Poisson distribution if it counts the number of events that occur at random in a given time or space

• For a Poisson distribution to be valid it must satisfy the following properties:

  • Events occur singly and at random in a given interval of time or space

  • The mean number of occurrences in the given interval(λ) is known and finite

    • λ has to be positive but does not have to be an integer

  • Each occurrence is independent of the other occurrences

• If X follows a Poisson distribution then it is denoted $X \text{tilde} Po ( \lambda )$

  • λ is the mean number of occurrences of the event

• The formula for the probability of r occurrences in a given interval is:

  • $P(X=r)=e^{-λ}\times\frac{λ^{r}}{r!}$ for r=0, 1, 2, ...,n
  • e is the constant 2.718…
  • r factorial=r (r−1)(r−2)....2×1

What are the important properties of a Poisson distribution?

• The mean and variance of a Poisson distribution are roughly equal

• The distribution can be represented visually using a vertical line graph

  • If λ is close to 0 then the graph has a tail to the right (positive skew)
  • If λ is at least 5 then the graph is roughly symmetrical

• The Poisson distribution becomes more symmetrical as the value of the mean (λ) increases

Worked Example $X$ is the random variable ‘The number of cars that pass a traffic camera per day’. State the conditions that would need to be met for $X$ to follow a Poisson distribution.

Answer:

Modelling with Poisson Distribution

Spec point — 1.1

How do I set up a Poisson model?

• Find the mean and variance and check that they are roughly equal

  • You may have to change the mean depending on the given time/space interval

• Make sure you clearly state what your random variable is

  • For example, let X be the number of typing errors per page in an academic article

• Identify what probability you are looking for

What can be modelled using a Poisson distribution?

• Anything that occurs singly and randomly in a given interval of time or space and satisfies the conditions

• For example, let X be the random variable 'the number of emails that arrive into your inbox per day'

  • There is a given interval of a day, this is an example of an interval of time

  • We can assume the emails arrive into your inbox at random

  • We can assume each email is independent of the other emails

    • This is something that you would have to consider before using the Poisson distribution as a model

  • If you know the mean number of emails per day a Poisson distribution can be used

• Sometimes the given interval will be for space

  • For example, the number of daisies that exist on a square metre of grass
  • look carefully at the units given as you may have to change them when calculating probabilities

Worked Example State, with reasons, whether the following can be modelled using a Poisson distribution and if so write the distribution.

(i) Faults occur in a length of cloth at a mean rate of 2 per metre.

(ii) On average 4% of a certain population has green eyes.

(iii) An emergency service company receives, on average, 15 calls per hour.

Answer:

Exam Hint

• If you are asked to criticise a Poisson model always consider whether the trials are independent, this is usually the one that stops a variable from following a Poisson distribution!