Poisson sampling explained

Poisson sampling should not be confused with Poisson disk sampling.

In survey methodology, Poisson sampling (sometimes denoted as PO 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.^{[1] [2]}

Each element of the population may have a different probability of being included in the sample (

\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} ). 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 ith element of the population is denoted by the symbol

\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 \pi _{ij} .

The following relation is valid during Poisson sampling when

i ≠ j :\pi _{ij} = \pi _{i} x \pi _{j} . \pi _{ii} is defined to be \pi _{i} .

See also

• Bernoulli sampling
• Poisson distribution
• Poisson process
• Sampling design

Notes and References

1. Book: Model Assisted Survey Sampling . Carl-Erik Sarndal . Bengt Swensson . Jan Wretman . 978-0-387-97528-3 . 1992.
2. 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)