# Poisson sampling

In the theory of finite population sampling, **Poisson sampling** is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample.

Each element of the population may have a different probability of being included in the sample. The probability of being included in a sample during the drawing of a single sample is denoted as the *first-order inclusion probability* of that element. If all first-order inclusion probabilities are equal, Poisson sampling becomes equivalent to Bernoulli sampling, which can therefore be considered to be a special case of Poisson sampling.

## A mathematical consequence of Poisson sampling

Mathematically, the first-order inclusion probability of the *i*th element of the population is denoted by the symbol π_{i} and the second-order inclusion probability that a pair consisting of the *i*th and *j*th element of the population that is sampled is included in a sample during the drawing of a single sample is denoted by π_{ij}.

The following relation is valid during Poisson sampling:

## Further reading

- Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, ISBN 0-387-40620-4