People, businesses, and organizations study probability theories so they can take better advantage of opportunities as well as manage uncertainties. The mathematical study of probability provides them a way of assigning a value to possible outcomes. Since their resources are finite, they can make informed decisions where to prioritize and how much to prepare for the outcomes most likely to happen. So probability models help companies optimize their policies and processes to ensure business success.
Poisson distribution
Poisson distribution is a class of a discrete probability distribution where the set of possible outcomes is discrete, distinct, or independent. For example, in an experiment where someone tosses a fair coin every six seconds for one minute, the possible outcomes are distinctly either heads or tails, and can never be both or neither.
We can say that this experiment has a Poisson distribution where the probability value of the coin coming up as heads is 0.5 or 50 percent. The same probability value can be assigned to the coin coming up as tails.
Assumptions and conditions
For a Poisson distribution to be valid and applicable, certain assumptions and conditions must be met.
- An event can occur any number of times during a time period, where the number of occurrences theoretically can range from zero to infinity.
- Events occur independently from other events.
- Expected number or rate of occurrence is assumed to be constant throughout the experiment.
- The probability of an event occurring is proportional to the length of the time period.
- Events produce discrete outcomes.
Based on these assumptions and conditions, some events are not good candidates for this statistical tool. For example, you cannot use the distribution to show the number of days patients spend in the ICU because it will never be zero. You cannot use it also to show the number of earthquakes of a certain magnitude occurring per year, because a stronger earthquake can trigger an aftershock with a magnitude that you’re similarly monitoring. Therefore, the event did not occur independently. The tool cannot also describe accurately the number of students arriving per minute at a gathering, because the rate will be lower during class time, and higher during break time. So the rate of occurrence is not constant.
Use cases
On the other hand, the Poisson distribution has been applied to model many events.
- Telecommunications/management: The number of phone calls received at a call center within a minute.
- Staffing/scheduling: The optimal number of agents to handle customer service calls during peak hours.
- Transportation/operations: The number of cars randomly arriving at a car wash in a 20-minute period.
- Sales/marketing: The number of customers per minute entering a bookstore.
- Sports/gaming: The number of goals scored in a World Cup game.
- Biology: The number of mutations on a strand of DNA after a limited exposure to radiation.
Other applications of the tool include astronomy, chemistry, finance, insurance, seismology, radioactivity, optics, and medicine.
Businesses can use probability studies, for instance, in their calculation when making investment decisions. They can use it to solve or prevent loss of customers. Others use it to develop competitive strategies that account for uncertainties or determine the probability of failure or breakdown of a product to create a better or more efficient design.
Project management use cases for Poisson distribution
Project management teams can use Poisson distribution during its planning phase to gather project requirements.
For example, a company with limited financial resources wants to determine if they need to upgrade or acquire an additional machine that is averaging a downtime of one hour every month (rate of occurrence or é=1) for the past year. So far, the downtime seems to be within the acceptable limits. But they want to know what the probability of the downtime increasing to two hours per month. It should not be more than 15 percent according to their policy.
Poisson distribution formula, where average é=1, e = Euler’s number, X is index of 2 hours of downtime :
P(X=2) = ( éX * e– é )/(X!)
P(X=2) = (1 * 0.3679)/(2)
P(x=2) = 0.1839
Therefore, there is a 18.39 percent probability of a downtime of two hours per month, which is more than their 15 percent threshold. Therefore, the project team can recommend to upgrade or add another machine, which should be included in their project plan.
Conclusion
The Poisson distribution is a useful statistical tool to use when you want to improve the operations of a business, to get an overview of probability values to help with project requirements and planning, and to describe other rare or discrete events.
