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Concept
Hypergeometric distribution
Summary
In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of successes in draws with replacement.
The following conditions characterize the hypergeometric distribution:
The result of each draw (the elements of the population being sampled) can be classified into one of two mutually exclusive categories (e.g. Pass/Fail or Employed/Unemployed).
The probability of a success changes on each draw, as each draw decreases the population (sampling without replacement from a finite population).
A random variable follows the hypergeometric distribution if its probability mass function (pmf) is given by
where
is the population size,
is the number of success states in the population,
is the number of draws (i.e. quantity drawn in each trial),
is the number of observed successes,
is a binomial coefficient.
The () is positive when .
A random variable distributed hypergeometrically with parameters , and is written and has probability mass function above.
As required, we have
which essentially follows from Vandermonde's identity from combinatorics.
Also note that
This identity can be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter, but it
also follows from the symmetry of the problem. Indeed, consider two rounds of drawing without replacement. In the first round, out of neutral marbles are drawn from an urn without replacement and coloured green. Then the colored marbles are put back. In the second round, marbles are drawn without replacement and colored red. Then, the number of marbles with both colors on them (that is, the number of marbles that have been drawn twice) has the hypergeometric distribution.
Official source
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