When to use Poisson distribution

This article is a continuation from the previous Poisson distribution article It may be a good idea to go through it before continuing with this article. This article covers how to decide whether or not a Poisson distribution is a suitable model. In Poisson distribution the random variable X represents the number of events that occur in an interval. The interval may be a fixed length in space or time. If X is to have a Poisson distribution then the events must occur;

• Singly in space or time
• independently of each other
• at a constant rate in the sense that the mean number of occurrences in the interval is proportional to the length of the interval.

The events must occur randomly . Below are some examples;

Example

Some river water contains on average 500 microbes per little. A large bucket of water is collected and after it has been well stirred a 1cm³ sample is examined.
The importance of stirring is to avoid the possible problem of the microbes occurring in clusters. We can relate this to the condition for Poisson distribution which states events must occur singly/independently in space or time.
Suppose we had to find the probability of there being no microbes in the sample.
We know that the microbes occur at a rate of 500 per litre. To use a Poisson distribution we will need the average in 1cm³. We let λ equal to the average number of microbes in 1cm³.

Let X = the number of microbes in 1cm³.

Suppose we had to find the probability of there being at least 4 microbes in the sample. We form an expression as;

P(X ≥ 4) is the same as 1 – P(X ≤ 3), so we have;

For P(X ≤ 3) we look in the tables of Poisson distribution to find where x = 3, and λ = 0.5, so we get;

The probability of there being at least 4 microbes in the sample is 0.0018.

Example

Here is another example. A shop sells radios at a rate of 2.5 per week.
Suppose we had to find the probability that in a two-week period the shop sells at least 7 radios.
We let X = the number of radios sold in two weeks.
The hint here in whether a Poisson distribution can be used comes from the use of the word rate. The rate given is a rate of 2.5 per week, however we’re trying to find the probability in a two-week period so we need to find the mean for two weeks. That is;

…so now we have the distribution as;

We’re trying to find P(X ≥ 7) and we know that;

…so we get;

Suppose we wanted to find the probability of selling fewer than 12 radios in a four-week period.
We need to use a new random variable to represent the number sold in 4 weeks. We let Y = the number of radios sold in a four-week period. So we have a distribution of;

…we’re looking for;

…and we know that;

So now we can use the tables of Poisson distribution to find where λ = 10 and x = 11, we get;