# 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
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 Resources

### Mr Mathematics Blog

#### Estimating Solutions by Rounding to a Significant Figure

Explore key concepts, FAQs, and applications of estimating solutions for Key Stage 3, GCSE and IGCSE mathematics.

#### Understanding Equivalent Fractions

Explore key concepts, FAQs, and applications of equivalent fractions in Key Stage 3 mathematics.

#### Transforming Graphs Using Function Notation

Guide for teaching how to transform graphs using function notation for A-Level mathematics.