Poisson Trial

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.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
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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 ''i''th element of the population is denoted by the symbol $\pi_i$ 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 $\pi_$.

The following relation is valid during Poisson sampling when $i\neq j$:

:$\pi_ = \pi_ \times \pi_.$

$\pi_$ is defined to be $\pi_i$.

See also

* Bernoulli sampling
* Poisson distribution
* Poisson process
* Sampling design

References

Sampling techniques

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