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?

• A binomial distribution $X~\mathrm{B}\left(n,p\right)$ can be approximated by a Poisson distribution $X_p~\mathrm{Po}\left(\lambda \right)$ 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 $\mathit{n}\mathit{p}$ should be $\mathbf{\le }\mathbf{10}$

• The mean to use in the approximation can be calculated by:

  • $\mathit{\lambda }\mathbf{=}\mathit{n}\mathit{p}$

  • This gives the Poisson the same mean as the binomial

  • Recall that for the binomial distribution

    • the mean is $np$

    • the variance is $np\left(1-p\right)$

• If $n$ is large but $p$ is near to 1, consider modelling the number of failures, $X\text{'}$

  • $X\text{'}~B\left(n,1-p\right)$

    • $1-p$ 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