Editing Hypergeometric distribution (section)

=== Multivariate hypergeometric distribution ===
{{Infobox probability distribution
| name       = Multivariate hypergeometric distribution
| type       = mass
| pdf_image  =
| cdf_image  =
| parameters = <math>c \in \mathbb{N}_{+} = \lbrace 1, 2, \ldots \rbrace</math><br />
<math>(K_1,\ldots,K_c) \in \mathbb{N}^c</math><br />
<math>N = \sum_{i=1}^c K_i</math><br /><math>n \in \lbrace 0,\ldots,N\rbrace</math>
| support    = <math>\left\{ \mathbf{k} \in \left(\mathbb{Z}_{0+}\right)^c \, : \, \forall i\ k_i \le K_i , \sum_{i=1}^{c} k_i = n \right\}</math>
| pdf        = <math>\frac{\prod\limits_{i=1}^c \binom{K_i}{k_i}}{\binom{N}{n}}</math>
| cdf        =
| mean       = <math>\operatorname E(k_i) = n\frac{K_i}{N}</math>
| median     =
| mode       =
| variance   = <math>\operatorname{Var}(k_i) = n \frac{N-n}{N-1} \;\frac{K_i}{N} \left(1-\frac{K_i}{N}\right) </math><br />
<math>\operatorname{Cov}(k_i,k_j) = -n \frac{N-n}{N-1} \;\frac{K_i}{N} \frac{K_j}{N}, i\ne j</math><br /><math>\operatorname{Corr}(k_i,k_j) = -\sqrt{\frac{K_i K_j}{\left(N-K_i\right)\left(N-K_j\right)}}</math>
| skewness   =
| kurtosis   =
| entropy    =
| mgf        =
| char       =
}}

The model of an [[urn problem|urn]] with green and red marbles can be extended to the case where there are more than two colors of marbles. If there are ''K''<sub>''i''</sub> marbles of color ''i'' in the urn and you take ''n'' marbles at random without replacement, then the number of marbles of each color in the sample (''k''<sub>1</sub>, ''k''<sub>2</sub>,..., ''k''<sub>''c''</sub>) has the multivariate hypergeometric distribution:  

:<math>\Pr(X_1 = k_1, \ldots, X_c = k_c) = \frac{\prod\limits_{i=1}^c \binom{K_i}{k_i}}{\binom{N}{n}}</math>

This has the same relationship to the [[multinomial distribution]] that the hypergeometric distribution has to the binomial distribution—the multinomial distribution is the "with-replacement" distribution and the multivariate hypergeometric is the "without-replacement" distribution.

The properties of this distribution are given in the adjacent table,<ref>{{cite arXiv |last=Duan |first=X. G. |title=Better understanding of the multivariate hypergeometric distribution with implications in design-based survey sampling |eprint=2101.00548 |date=2021 |class=math.ST }}</ref> where ''c'' is the number of different colors and <math>N=\sum_{i=1}^c K_i</math> is the total number of marbles in the urn.

==== Example ====
Suppose there are 5 black, 10 white, and 15 red marbles in an urn.  If six marbles are chosen without replacement, the probability that exactly two of each color are chosen is
: <math> P(2\text{ black}, 2\text{ white}, 2\text{ red}) = {{{5 \choose 2}{10 \choose 2} {15 \choose 2}}\over {30 \choose 6}} = 0.079575596816976</math>