Poisson Distribution:

Introduction:

The Poisson distribution is a special case of the binomial distribution that it models discrete events. It expresses the probability of a number of relatively rare events occurring in a fixed time if these events occur with a known average rate and are independent of the time since the last event.

In summary, the conditions under which a Poisson distribution holds are when:

• counts are of rare events.
• all events are independent.
• average rate does not change over the period of interest.

In physics, these conditions normally occur when we are dealing with counting – especially radioactive decay, or photon counting using a Geiger tube.

Unlike the normal or binomial distributions, the only parameter we need to define is the average rate, or the mean of the distribution, for which N̅, or λ, are often used. The Poisson probability distribution is a single-parameter function which allows us to find the probability that there are exactly x occurrences (x being a non-negative integer, x = 0, 1, 2, ...). It is defined as 

where N̅ is a positive number (not necessarily an integer) equal to the mean. As in the case for the Gaussian distribution we can find the mean and the standard deviation of the Poisson probability function. The average count rate, or mean, is N̅. As the function is only defined by one variable, it may not be surprising to find that the standard deviation is also related to the mean. The standard deviation of a Poisson distribution is simply the square root of the mean! 

Fig. 1: The evolution of the Poisson distribution as the mean increases. The Gaussian distribution as a function of the continuous variable x is superimposed.

Standard Error

The standard error for the Poisson distribution is the square root of the number of counts, i.e 

Using Excel:

Excel can make calculations using the Poisson distribution equation for us. To illustrate this, let’s use an example. This example was taken from “Measurements and their Uncertainties”, I. Hughes and T. Hase.

A safety procedure at a nuclear power plant stops the nuclear reactions in the core if the background radiation level exceeds 13 counts per minute. In a random sample, the total number of counts recorded in 10 hours was 1980. What is the count rate per minute and its error? What is the probability that during a random one-minute interval 13 counts will be recorded? What is the probability that the safety system will trip?

We are going to use this example to show how excel can be used to make this problem simpler. A spreadsheet has already been prepared for you Excel Guide Template and answers (see Distributions pdf ) are available in case you get stuck!

So why not use a calculator instead?

There are two main reasons for this:

1. The cumulative Poisson calculation would have been very lengthy.
2. If you want to change any of the starting numbers, the resulting calculations will change automatically. You don't have to start from scratch!

The second point is the most important. Regularly in your experiments you'll find that you'll only have to change one or two starting variables to a calculation, setting up a spreadsheet to do this easily saves a lot of time!