In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the
population
Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
is subjected to an
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
Bernoulli trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is c ...
which determines whether the element becomes part of the sample. An essential property of Bernoulli sampling is that all elements of the population have equal probability of being included in the sample.
Bernoulli sampling is therefore a special case of
Poisson sampling. In
Poisson sampling each element of the population may have a different probability of being included in the sample. In Bernoulli sampling, the probability is equal for all the elements.
Because each element of the population is considered separately for the sample, the sample size is not fixed but rather follows a
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...
.
Example
The most basic Bernoulli method generates ''n'' random variates to extract a sample from a population of ''n'' items. Suppose you want to extract a given percentage ''pct'' of the population. The algorithm can be described as follows:
for each item in the set
generate a random non-negative integer R
if (R mod 100) < pct then
select item
A percentage of 20%, say, is usually expressed as a probability ''p''=0.2. In that case, random variates are generated in the unit interval. After running the algorithm, a sample of size ''k'' will have been selected. One would expect to have
, which is more and more likely as ''n'' grows. In fact, It is possible to calculate the probability of obtaining a sample size of ''k'' by the
Binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...
:
On the left this function is shown for four values of
and
. In order to compare the values for different values of
, the
's in abscissa are scaled from