Generalized hypergeometric distribution

The multivariate hypergeometric distribution , also called generalized hypergeometric distribution , general hypergeometric distribution or polyhypergeometric distribution , is a multivariate probability distribution and is one of the discrete probability distributions . It is a multivariate generalization of the hypergeometric distribution and can be derived from the urn model .

definition

A random variable with values ​​in is called multivariate hypergeometrically distributed to the parameters with and , if they are the probability function ${\displaystyle X}$ ${\displaystyle \{(b_{1},\ldots ,b_{k})\in \mathbb {N} _{0}^{k}\,|\,b_{1}+\ldots +b_{k}=n\}}$ ${\displaystyle B=(B_{1},\ldots ,B_{k})\in \mathbb {N} _{0}^{k}}$ ${\displaystyle B_{1}+\ldots +B_{k}=N}$ ${\displaystyle n\leq N}$

${\displaystyle f_{B,n}(b_{1},\dots ,b_{k})={\frac {{\binom {B_{1}}{b_{1}}}\cdot {\binom {B_{2}}{b_{2}}}\cdot \ldots \cdot {\binom {B_{k}}{b_{k}}}}{\binom {N}{n}}}}$

owns. Then you write or as with the hypergeometric distribution. ${\displaystyle X\sim {\mathcal {H}}_{B,n}}$ ${\displaystyle X\sim Hyp_{B,n}}$

Derivation from the urn model

The multivariate hypergeometric distribution can be clearly derived from the urn model. Consider an urn with a total of balls, each of which is colored in a different color. There are balls of the same color . The probability of pulling balls of the same color when pulling them once without replacing them is multivariate hypergeometrically distributed. ${\displaystyle N}$ ${\displaystyle k}$ ${\displaystyle i}$ ${\displaystyle B_{i}}$ ${\displaystyle n}$ ${\displaystyle b_{i}}$ ${\displaystyle i}$

properties

Expected value

If the number of balls is the same color , then is the expected value ${\displaystyle X_{i}}$ ${\displaystyle i}$

${\displaystyle \operatorname {E} (X_{i})={\frac {nB_{i}}{N}}}$

Variance

The variance is

${\displaystyle \operatorname {Var} (X_{i})={\frac {B_{i}}{N}}\left(1-{\frac {B_{i}}{N}}\right)n{\frac {N-n}{N-1}}}$

Covariance

The following applies for the covariance between the number of balls

${\displaystyle \operatorname {Cov} (X_{i},X_{j})=-{\frac {nB_{i}B_{j}}{N^{2}}}{\frac {N-n}{N-1}}}$

if . ${\displaystyle i\neq j}$

example

There is an urn with 5 black, 10 white and 15 red balls. The probability of drawing exactly two balls of each color when drawing six times is

${\displaystyle P(2{\text{ schwarz}},2{\text{ weiss}},2{\text{ rot}})={{{5 \choose 2}{10 \choose 2}{15 \choose 2}} \over {30 \choose 6}}\approx 0.0795}$,

so just under eight percent. It is . This follows, for example, for the expected value of the black balls . ${\displaystyle n=6,N=30,B_{1}=5,B_{2}=10,B_{3}=15}$ ${\displaystyle \operatorname {E} ({\text{schwarz}})=1}$

Relationship to other distributions

Relationship to the hypergeometric distribution

The hypergeometric distribution is a special case of the multivariate hypergeometric distribution with and . Please note the different parameterizations here. ${\displaystyle B_{1}=M}$ ${\displaystyle B_{2}=N-M}$

Relationship to the multinomial distribution

The multivariate hypergeometric distribution and the multinomial distribution are related because they arise from the same urn model, with the difference that it is covered in the multinomial model. In particular, it can be shown that if and holds such that is, and which define a probability function on , then converges point by point to the multinomial distribution with the parameters and . The multivariate hypergeometric distribution can thus be approximated by the multinomial distribution. ${\displaystyle N\rightarrow \infty }$ ${\displaystyle B_{i}\rightarrow \infty }$ ${\displaystyle {\tfrac {B_{i}}{N}}\rightarrow p_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle \{1,\dots ,k\}}$ ${\displaystyle {\mathcal {H}}_{B,n}}$ ${\displaystyle {\mathcal {M}}_{p,n}}$ ${\displaystyle p_{i}}$ ${\displaystyle n}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .