Multinomial distribution

The multinomial distribution or polynomial distribution is a probability distribution in stochastics . It is a discrete probability distribution and can be understood as a multivariate generalization of the binomial distribution . In Bayesian statistics, it has the Dirichlet distribution as conjugate a priori distribution .

Definition and model

Be and with . Then the counting density of the multinomial distribution is given by ${\displaystyle n,k\in \mathbb {N} _{0}}$ ${\displaystyle p_{1},\dotsc ,p_{k}\in [0,1]}$ ${\displaystyle p_{1}+\dotsb +p_{k}=1}$ ${\displaystyle M(n,(p_{1},\dotsc ,p_{k}))}$

${\displaystyle f(n_{1},\dotsc ,n_{k})\;=\;{\begin{cases}{n \choose n_{1},\dotsc ,n_{k}}\;p_{1}^{n_{1}}\dotsm p_{k}^{n_{k}}{\text{,}}&{\text{wenn }}n_{1},\dotsc ,n_{k}\in \mathbb {N} _{0}{\text{ und }}n_{1}+\dotsb +n_{k}=n\\0&{\text{sonst.}}\end{cases}}}$.

Here is the multinomial coefficient . ${\displaystyle {n \choose n_{1},\dotsc ,n_{k}}={\frac {n!}{n_{1}!\dotsm n_{k}!}}}$

Application and motivation

The multinomial distribution can be motivated based on an urn model with replacement. In an urn there are sorts of balls. The proportion of varieties balls in the urn is . One ball is removed from the urn , its property (type) is noted and the ball is then returned to the urn. ${\displaystyle k}$ ${\displaystyle p_{i},(i\in \{1,\dotsc ,k\})}$ ${\displaystyle n}$

One is now interested in the number of spheres of each type in this sample. Since the multinomial distribution follows, the sample has the probability: ${\displaystyle x_{i}}$ ${\displaystyle i}$ ${\displaystyle (X_{1},\dotsc ,X_{k})}$ ${\displaystyle x_{1},\dotsc ,x_{k}}$

${\displaystyle P(X_{1}=x_{1},X_{2}=x_{2},\dotsc ,X_{k}=x_{k})={\frac {n!}{x_{1}!\cdot x_{2}!\dotsm x_{k}!}}p_{1}^{x_{1}}\cdot p_{2}^{x_{2}}\dotsm p_{k}^{x_{k}}}$.

Taking an urn containing varieties balls, each with a ball per variety, we get the classic dice: You throw these times, it is possible outputs and interested in how big is the probability that straight times occurs just times and so on. Furthermore, the respective descriptions describe the probabilities of the faces of the cube and thus whether it is a fair or an unfair cube. ${\displaystyle k=6}$ ${\displaystyle n}$ ${\displaystyle k=6}$ ${\displaystyle X_{1}}$ ${\displaystyle x_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle x_{2}}$ ${\displaystyle p}$

properties

Expected value

For each the random variable is binomially distributed with the parameters and , thus has the expected value ${\displaystyle i}$ ${\displaystyle X_{i}}$ ${\displaystyle n}$ ${\displaystyle p_{i}}$

${\displaystyle \operatorname {E} (X_{i})=np_{i}}$

Variance

The following applies to the variance

${\displaystyle \operatorname {Var} (X_{i})=np_{i}(1-p_{i})}$.

Covariance and Correlation Coefficient

The covariance of two random variables and with is calculated as ${\displaystyle X_{i}}$ ${\displaystyle X_{j}}$ ${\displaystyle i\neq j}$

${\displaystyle \operatorname {Cov} (X_{i},X_{j})=-np_{i}p_{j}}$,

and for the correlation coefficient (according to Pearson) it follows:

${\displaystyle \varrho (X_{i},X_{j})=-{\sqrt {{\frac {p_{i}}{1-p_{i}}}{\frac {p_{j}}{1-p_{j}}}}}}$.

Probability generating function

The multivariate probability generating function is

${\displaystyle m_{X}(t)=\left(\sum _{i=1}^{k}p_{i}t_{i}\right)^{n}}$

example

There are 31 pupils in a school class, 12 from village A, 11 from village B and 8 from village C. Every day a pupil is drawn to wipe the blackboard. What is the probability that no student from village A, two students from village B and 3 students from village C have to wipe the blackboard in one week? It is and , since every student should be drawn equally likely. Then ${\displaystyle n_{\mathrm {A} }=0,n_{\mathrm {B} }=2,n_{\mathrm {C} }=3}$ ${\displaystyle p_{\mathrm {A} }={\frac {12}{31}},p_{\mathrm {B} }={\frac {11}{31}},p_{\mathrm {C} }={\frac {8}{31}}}$ ${\displaystyle M((0,2,3),p_{\mathrm {A} },p_{\mathrm {B} },p_{\mathrm {C} })={\frac {5!}{0!2!3!}}\left({\frac {11}{31}}\right)^{2}\left({\frac {8}{31}}\right)^{3}={\frac {10\cdot 11^{2}\cdot 8^{3}}{31^{5}}}\approx 0{,}022}$

Relationship to other distributions

Relationship to the binomial distribution

In the special case , the binomial distribution results , more precisely is the joint distribution of and for a -distributed random variable . ${\displaystyle k=2}$ ${\displaystyle M(n,(p,1-p))}$ ${\displaystyle X}$ ${\displaystyle n-X}$ ${\displaystyle B(n,p)}$ ${\displaystyle X}$

Relationship to the multivariate hypergeometric distribution

The multinomial distribution and the multivariate hypergeometric distribution are related because they come from the same urn model. The only difference is that the multivariate hypergeometric distribution draws without replacement. The multivariate hypergeometric distribution can be approximated under certain circumstances by the multinomial distribution, see the article on the multivariate hypergeometric distribution.

literature

• Ulrich Krengel : Introduction to probability theory and statistics . 8th edition, Vieweg, 2005. ISBN 978-3-834-80063-3
• Hans-Otto Georgii: Stochastics: Introduction to Probability Theory and Statistics , 4th edition, de Gruyter, 2009. ISBN 978-3-110-21526-7
• Christian Hesse: Applied probability theory : a well-founded introduction with over 500 realistic examples and tasks, Vieweg, Braunschweig / Wiesbaden 2003, ISBN 978-3-528-03183-1 .