A count of successes has a Poisson distribution in the Poisson setting: the number of successes that occur in two nonoverlapping units of measure are independent; the probability that a success will occur in a unit of measure is the same for all units of equal size and is proportional to the size of the unit; the probability that more than one event occurs in a unit of measure is negligible for very small sized units. In other words, the events occur one at a time.
If has the Poisson distribution with mean , then the standard deviation of is , and the possible values of are the whole numbers 0, 1, 2, 3, and so on.
The Poisson probability that takes any of these values is
Sums of independent Poisson random variables also have the Poisson distribution. For example, in a Poisson model with mean per unit of measure, the count of successes in units is a Poisson random variable with mean .
A simple check for the adequacy of the Poisson model is to compare the closeness of the observed mean count with the observed variance of the counts. In addition, some software packages provide fitting of the Poisson model on the observed histogram to assess compatibility.