It is used to model count-based data, like the number of emails arriving in your mailbox in one hour or the number of customers walking into a shop in one day, for instance. It can be used to predict how many times an event might occur in a given time period.
The number (k) of hits on a website in one hour with an average hit rate of 6 hits per hour is poisson distributed.
Insurance companies to conduct risk analysis (eg. predict the number of car crash accidents within a predefined time span) to decide car insurance pricing.
The Poisson distribution is parametrized by the expected number of events λ (pronounced “lambda”) in a time or space window. The distribution is a function that takes the number of occurrences of the event as input (the integer called k in the next formula) and outputs the corresponding probability (the probability that there are k events occurring).
for k = 0, 1, 2, ... The formula of P(k; λ) returns the probability of observing k events given the parameter λ which corresponds to the expected (sometimes called average) number of occurrences in that time slot.