Poisson Approximations of Binomials

When can I use a Poisson distribution to approximate a binomial distribution?

A binomial distribution $X ~ B (n, p)$ can be approximated by a Poisson distribution $X_{p} ~ Po (λ)$ provided
- n is large
- p is small
- There is no firm rule for what 'large' and 'small' mean here
  - $n > 30$ is a good guide for 'large $n$ '
  - usually the value of $n p$ should be $\leq 10$
The mean to use in the approximation can be calculated by:
- $λ = n p$
- This gives the Poisson the same mean as the binomial
- Recall that for the binomial distribution
  - the mean is $n p$
  - the variance is $n p (1 - p)$
If $n$ is large but $p$ is near to 1, consider modelling the number of failures, $X'$
- $X' ~ B (n, 1 - p)$
  - $1 - p$ will be small
  - A Poisson approximation can then be used
The Poisson distribution is derived from the binomial distribution by letting n become infinitely large and p become infinitely small

Exam Tip

An exam question will generally state if you need to use a Poisson approximation

Worked example

It is known that one person in a thousand who checks a revision website will choose to subscribe. Given that the website received 3000 hits yesterday, use a Poisson approximation to find the probability that more than 5 people subscribed.

1-5-3-poisson-approx-of-binomial-we-solution

Poisson Approximations of Binomials (Edexcel A Level Further Maths: Further Statistics 1)

Revision Note

Poisson Approximations of Binomials

When can I use a Poisson distribution to approximate a binomial distribution?

Exam Tip

Worked example

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Author: Roger