What distribution is used when occurrences of an event are small but the number of cases is large enough to ensure a few occurrences?

The Poisson distribution is particularly suited for modeling the number of times an event occurs in a fixed interval of time or space when these events are rare relative to the total number of opportunities for their occurrence. This distribution is characterized by its focus on events that happen independently of one another and where the average rate of occurrence is constant.

In practical applications, the Poisson distribution assumes that while occurrences may be infrequent (small number of events), the range of possible cases or units where these events may occur is sufficiently large. For instance, in safety management, one might use the Poisson distribution to model the number of worker injuries happening within a large workforce over a specific time frame, where injuries are relatively rare but the total population is numerous.

The other distributions mentioned serve different purposes. The normal distribution is more frequently applied to situations involving continuous data that clusters around a mean. The binomial distribution applies to scenarios with a fixed number of trials where each trial results in a binary outcome (success or failure). The exponential distribution is used to model the time until the next event occurs in a continuous process where events happen independently at a constant average rate. This gives an idea of how the Poisson distribution specifically applies to the scenario of rare event occurrences within a large population