Poisson Probability Distribution

The Poisson distribution models discrete events occurring over a specified interval such as time, volume, or distance. It applies when events happen at a known average rate and independently of each other. Examples include daily car sales, misspelled words in a book, or any countable occurrences within a fixed interval where the average rate is known.

Unlike the binomial probability distribution, the Poisson distribution depends only on the mean and is unaffected by sample size or probability of individual events. The binomial distribution requires a fixed number of trials and constant probability, while Poisson applies to events in a continuous interval with no upper limit on the random variable.

The standard deviation of a Poisson distribution equals the square root of the mean. This relationship is unique to Poisson distributions and simplifies calculations since you need only the mean value to determine both the distribution's center and spread around that mean.

Yes, the Poisson distribution can approximate the binomial probability distribution when the probability of success is small (such as 0.01) and the number of trials is large (such as 1,000). This approximation simplifies calculations when binomial conditions are met but parameters make direct computation difficult.

The Poisson distribution models independent events occurring at a constant average rate within a fixed interval. Since each event is independent and the rate is known, only the mean is needed to describe the probability of any number of occurrences. Previous event history does not influence future probabilities in this model.

The Poisson formula calculates the probability of exactly k events occurring given the mean. For example, with an average of three daily car sales, you can calculate the probability of selling exactly four cars on a given day. Once calculated, all other probabilities can be plotted in probability histograms for visual representation.

Poisson distribution applies to countable events in fixed intervals where the average rate is known and events are independent. Examples include the number of misspelled words in 100 pages of a book, customer arrivals per hour at a service desk, or defects per unit of production. Any scenario with a predictable average rate over a defined interval fits this model.