Polynomial Experiments and Contingency Tables
Definition 1
Multinomial Experiment
An experiment that has the following characteristics and has three or more possible outcomes or categories is called a Multinomial Experiment.
- It consists of $n$ identical trials.
- Each trial’s outcome is one of $k>2$ possible outcomes or categories.
- Each trial is independent.
- The probabilities of various outcomes remain constant throughout the trials.
Contingency Table
When there is information about more than one variable for an element, the table summarizing this is called a Contingency Table.
| X | Y | |
|---|---|---|
| Aaa | 0000 | 0000 |
| Bbb | 0000 | 0000 |
| Ccc | 0000 | 0000 |
Description
A Multinomial Experiment is simply an experiment to obtain data when assuming a multinomial distribution.
Multinomial Distribution: Let us denote a random vector composed of $n \in \mathbb{N}$ and categories $k \in \mathbb{N}$ amount of random variables as $\left( X_{1} , \cdots , X_{k} \right)$. $$ \sum_{i=1}^{k} X_{i} = n \qquad \& \qquad \sum_{i=1}^{k} p_{i} = 1 $$ For $\mathbf{p} = \left( p_{1} , \cdots , p_{k} \right) \in [0,1]^{k}$ that satisfies, the following probability mass function of multivariate probability distribution $M_{k} \left( n, \mathbf{p} \right)$ is called a Multinomial Distribution. $$ p \left( x_{1} , \cdots , x_{k} \right) = {{ n! } \over { x_{1} ! \cdots x_{k}! }} p_{1}^{x_{1}} \cdots p_{k}^{x_{k}} \qquad , x_{1} , \cdots , x_{k} \in \mathbb{N}_{0} $$
경북대학교 통계학과. (2008). 엑셀을 이용한 통계학: p266, 270. ↩︎