Notes on using PoiBinNorm.xls

This spreadsheet allows students to explore the behaviour of and relationship between the Poisson, Binomial and Normal distributions. The values of n and p for the Binomial can be set. The Poisson distribution is constrained to have the same mean as the Binomial by using the relationship mean µ = np unless the option of looking at the Poisson and Normal only is chosen.

The sheet can be used to introduce the idea of the various approximations and the circumstances in which they do or do not work well.

Background: The theory of Discrete Random Variables, the Binomial and Poisson distributions and their means and variances.The Normal distribution.

Questions for students:

• Set the mean equal to 1 and the value of n to its minimum. What important differences do you notice between the two distributions. Now gradually increase the value of n and observe what happens. For what values of n would you say that the Poisson distribution gave a good approximation to the Binomial? (Look at the table of values as well as the graph. You can use PgDn or the down arrow key to see more values.).

• Repeat the first question with some different values of n. Record your results (i.e. say for which values of n the fit seems good).

• The standard condition given by an exam board for the Poisson to be a good approximation to the Binomial is n > 50 and the mean np <5.

  • Do your results from the previous question agree with this?
  • Chose some values which fit this condition and see if the fit looks good.
  • Chose some values which do not fit this condition and see if the fit looks poor

• Set the mean and the value of n equal to their minimum values. Increase the value of the mean. What do you notice about the shapes of the distributions?