Poisson sampling

In survey methodology, Poisson sampling (sometimes denoted as PO sampling^[1]: 61) 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]: 85 [2]}

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

A mathematical consequence of Poisson sampling

Mathematically, the first-order inclusion probability of the ith element of the population is denoted by the symbol ${\displaystyle \pi _{i}}$ 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 ${\displaystyle \pi _{ij}}$.

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

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

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

See also

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

References