What's the difference between a binomial, hypergeometric, and Poisson probability distribution?

Question

Audrey Allen · Accepted Answer

Binomial - Random variable #X# is the number of successes in n independent and identical trials, where each trial has fixed probability of success. Hypergeometric - Random variable #X# is the number of objects that are special, among randomly selected #n# objects from a bag that contains a total of #N# out of which #K# are special. If #n# is much smaller than #N# then this can be approximated by binomial. Poisson - Random variable #X# counts the number of occurrences on an event in a given period, where we know that the concurrences has an average of #\lambda# for any period of that length, independent of any other disjoint period.

Robert Arnold · Answer

undefined The key differences between binomial, hypergeometric, and Poisson probability distributions lie in their underlying assumptions and the types of scenarios they model.

Binomial distribution: 
- Models the number of successes in a fixed number of independent Bernoulli trials.
- Assumes a fixed number of trials, each with two possible outcomes (success or failure).
- The probability of success remains constant for each trial.
- Example scenarios include coin flips, where the outcome of each flip is either heads or tails, or counting the number of defective items in a sample of fixed size from a production line.

Hypergeometric distribution:
- Models the number of successes in a sample drawn without replacement from a finite population.
- Assumes a fixed population size, a fixed number of successes in the population, and a fixed sample size.
- The probability of success changes as items are drawn from the population without replacement.
- Example scenarios include drawing cards from a deck without replacement, where the population consists of the total number of cards, and success is defined as drawing a specific type of card (e.g., aces or red cards).

Poisson distribution:
- Models the number of events occurring in a fixed interval of time or space.
- Assumes events occur independently and at a constant average rate.
- There is no upper limit on the number of events that can occur.
- Example scenarios include the number of phone calls received by a call center in an hour, the number of accidents at a particular intersection in a day, or the number of particles emitted by a radioactive source in a given time period.

In summary, the binomial distribution deals with a fixed number of trials with two possible outcomes, the hypergeometric distribution deals with sampling without replacement from a finite population, and the Poisson distribution deals with the number of events occurring in a fixed interval of time or space with a constant average rate.