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In survey methodology, Poisson sampling (sometimes denoted as PO samplingcite-ref-sarndal1992-1-0[1]) 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.cite-ref-sarndal1992-1-1[1]cite-ref-2[2]

Each element of the population may have a different probability of being included in the sample ( π π i {\displaystyle \pi _{i}} ). 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 ( p i {\displaystyle p_{i}} ). 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.

The name is derived from the fact that the number of samples approximate a Poisson distribution.

• A mathematical consequence of Poisson sampling

• See also

• References

Mathematically, the first-order inclusion probability of the ith element of the population is denoted by the symbol π π i {\displaystyle \pi _{i}} and the second-order inclusion probability that a pair consisting of the ith and jth element of the population that is sampled is included in a sample during the drawing of a single sample is denoted by π π i j {\displaystyle \pi _{ij}} .

The following relation is valid during Poisson sampling when i ≠ ≠ j {\displaystyle i\neq j} (i.e., Independence):

π π i j = π π i × × π π j . {\displaystyle \pi _{ij}=\pi _{i}\times \pi _{j}.}

π π i i {\displaystyle \pi _{ii}} is defined to be π π i {\displaystyle \pi _{i}} .

• Bernoulli sampling

• Poisson distribution

• Poisson process

• Sampling design

• Probability-proportional-to-size sampling

cite-note-sarndal1992-11. ↑ citerefcarl-erik-sarndalbengt-swenssonjan-wretman1992Carl-Erik Sarndal; Bengt Swensson; Jan Wretman (1992). Model Assisted Survey Sampling. ISBN 978-0-387-97528-3.

cite-note-22. ↑ Ghosh, Dhiren, and Andrew Vogt. "Sampling methods related to Bernoulli and Poisson Sampling." Proceedings of the Joint Statistical Meetings. American Statistical Association Alexandria, VA, 2002. (pdf)