The Poisson Distribution

Scheme of work: Year 12 Further Mathematics A-Level: Further Statistics 1: The Poisson Distribution

Throughout this unit, students learn to understand the Poisson distribution as a probability model and use it to solve real-life problems. Later, they use the Poisson distribution to approximate the binomial distribution as learning progresses.

Prerequisite Knowledge

  • Solve problems involving discrete random variables.
  • Calculate probabilities using the binomial distribution
  • Evaluate exponential expressions

Success Criteria

  • Understand the Poisson distribution as a probability model.
  • Calculate probabilities using the Poisson distribution
  • Understand and use the mean and variance of a Poisson distribution
  • Solve problems involving linear combinations of independent Poisson distributions
  • Use the Poisson distribution as an approximation to the binomial distribution

Key Concepts

  • A Poisson distribution can be used to model a discrete probability distribution in which the events occur:
    • independently
    • at random
    • at a constant average range in the given interval of space or time
    • singly, in space or time
  • The mean and variance of a Poisson distribution are equal. If the mean and variance are not approximately equal, the Poisson distribution is not a suitable model.
  • If the random variable, X has a Poisson distribution with parameter λ, where λ > 0, we write X ~ Po(λ) and
  • If X and Y are independent variables such that X ~ Po(λ) and Y~ Po(µ) then X + Y ~ Po(λ+µ)
  • If X~B(n,p), then for a large number of trials, n and small probability, p, then X~ Po(np)

Common Misconceptions

  • Students should be familiar with using the Poisson PD and CD functions on their Casio Classwiz as this will save time for questions later in the paper.
  • While most students understand when a Poisson model can be applied marks can be lost when explaining how it is used within the context of the question.
  • When asked to work out probabilities in the form P(X > a) students incorrectly calculate this as 1 – P( X < a) instead of 1 – P(X ≤ a)
  • Some students only work out the mean when asked to show why a Poisson distribution could be a suitable model for students. They need to show, often by working, that the mean and variance are approximately equal.
  • When asked to use a suitable approximation, for a binomial distribution students sometimes calculate the exact probability using a binomial model rather than converting to Poisson.

The Poisson Distribution Resources

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