📂Statistical Test

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.

1. It consists of $n$ identical trials.
2. Each trial’s outcome is one of $k>2$ possible outcomes or categories.
3. Each trial is independent.
4. 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. $$p \left( x_{1} , \cdots , x_{k} \right) = {{ n! } \over { x_{1} ! \cdots x_{k}! }} p_{1}^{x_{1}} \cdots p_{k}^{x_{k}} \qquad , x_{1} , \cdots , x_{k} \in \mathbb{N}_{0}$$