What is it?

The Poisson distribution (Haight, 1967) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

In other words, this distribution helps determine the probability of an event if it has been happening over and over again in some time interval.

The probability can be found using the following formula:

$$f(k; \lambda)= \Pr(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!}$$

where

• \(k\) is the number of occurrences (k = 0,1,2,3...)
• \(e\) is Euler's constant (2.71828...)
• \(\lambda\) is a real number > 0 that represents the event rate.

How can we use it?

So to exemplify this, let's create a hypothetical scenario. If we know that an earthquake hits San Francisco every 100 years one time (\(\lambda = 1\)), we might want to know what is the probability that \(k = {0,1,2,3}\) earthquakes can happen during the next 100 years.

$$f(k; \lambda)= \frac{\lambda^k e^{-\lambda}}{k!}$$

then:

$$f(0; 1)= \frac{1^0 e^{-1}}{0!} = 0.3678$$ $$f(1; 1)= \frac{1^1 e^{-1}}{1!} = 0.3678$$ $$f(2; 1)= \frac{1^2 e^{-1}}{2!} = 0.1839$$ $$f(3; 1)= \frac{1^3 e^{-1}}{3!} = 0.0613$$

This tells us that given that San Francisco has been shaking hard at least once every 100 years, the probability that no earthquake or at least one is the same, but the odds go down if we are expecting 2 or more earthquakes.

Assumptions

• All events (in our example earthquakes) are independent events. We assume one earthquake will not cause another one (at least from the same magnitude, its obvious replicas will happen).
• The rate of occurrence is also independent.
• The average rate is constant. Here we assume earthquakes happen every 100 years.
• Two events can't happen at the same instant. San Andreas Fault cant have two shakes at the same time.

Other questions answered by the Poisson Distribution:

• What is the possibility of selling 10 pizzas a day.
• What is the probability of Earth getting destroyed by a giant meteor.
• What is the probability that Cuba and the Dominican Republic will get hit by 3 tropical storms next year?
• What is the probability that Brazil will win the next world cup? (the WC happens every 4 years!)

References:

Haight, Frank A. (1967), Handbook of the Poisson Distribution, New York, NY, USA: John Wiley & Sons