The Poisson Distribution

July 22, 2022

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

E(X)=\sum_{x=0}^{\infty}{xP\left(X=x\right)=\lambda}

Var(X)=\sum_{x=0}^{\infty}{x^2P(X=x)-\lambda^2=\lambda}

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

July 22, 2022

Prerequisite Knowledge

Success Criteria

Key Concepts

Common Misconceptions

The Poisson Distribution Resources

Mr Mathematics Blog

Sequences and Series

T- Formulae

Regression, Correlation and Hypothesis Testing

The Poisson Distribution

July 22, 2022

Prerequisite Knowledge

Success Criteria

Key Concepts

Common Misconceptions

The Poisson Distribution Resources

Lesson

Mr Mathematics Blog

Sequences and Series

T- Formulae

Regression, Correlation and Hypothesis Testing