The binomial distribution, also known as Bernoulli distribution, describes the random sampling processes that all outcomes are either yes or no (success/failure) without ambiguity. In addition, the probability will not change from one trial to another (independent). In this case, the number of total available samples remains the same throughout the sampling process. In other words, the total number of available samples is either infinity or the chosen sample are always placed back to the sampling pool before the next trial (Sampling with replacement).

Suppose that the probability of success in a single trial is in a random sampling process and the failure rate is , where . The binomial distribution with exactly successes in trials, where , has the following important properties. See plots of binomial distributions.

Binomial Distribution
Density Function
Distribution Function
Mean
Variance
Standard Deviation

The hypergeometric distribution describes the random sampling processes that once a sample has been chosen, it will NOT be placed back to the sampling pool (sampling without replacement). In this case, the number of total available samples will decrease by one after each trial.

Suppose there are successes and failures in a sampling pool with available samples (). The hypergeometric distribution with exactly successes in trials, where , has the following important properties. See plots of hypergeometric distributions.

Hypergeometric Distribution
Density Function
Distribution Function
Mean
Variance
Standard Deviation

where and .

The Poisson distribution describes the random sampling process that the desired outcomes occur relatively infrequently but at a regular rate.

Suppose there are on average successes in a large number of trials (large sampling period). The Poisson distribution with exactly successes in the same sampling period has the following important properties. See plots of Poisson distributions.

Poisson Distribution
Density Function
Distribution Function
Mean
Variance
Standard Deviation

where .