Multinomial Distribution

In probability theory, the multinomial distribution is a generalization of the binomial distribution.The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial. In a multinomial distribution, each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p1, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables indicate the number of times outcome number i was observed over the n trials. follows a multinomial distribution with parameters n and p.
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The multinomial distribution arises when a response variable is categorical in nature, i.e., consists of data describing the membership of the respective cases to a particular category. For example, if a researcher recorded the outcome for the driver in accidents as "uninjured, "injury not requiring hospitalization", "injury requiring hospitalization", or "fatality", then the distribution of the counts in these categories would be multinomial (see Agresti, 1996). The multinomial distribution is a generalization of the binomial distribution to more than two categories.
If the categories for the response variable can be ordered, then the distribution of that variable is referred to as ordinal multinomial. For example, if in a survey the responses to a question are recorded such that respondents have to choose from the pre-arranged categories "Strongly agree", "Agree", "Neither agree nor disagree", "Disagree", and "Strongly disagree", then the counts (number of respondents) that endorsed the different categories would follow an ordinal multinomial distribution (since the response categories are ordered with respect to increasing degrees of disagreement).
Specialized methods for analyzing multinomial and ordinal multinomial response variables can be found in the Generalized Linear Models chapter.

If the categories for a multinomial response variable can be ordered, then the distribution of that variable is referred to as ordinal multinomial. For example, if in a survey the responses to a question are recorded such that respondents have to choose from the pre-arranged categories "Strongly agree", "Agree", "Neither agree nor disagree", "Disagree", and "Strongly disagree", then the counts (number of respondents) that endorsed the different categories would follow an ordinal multinomial distribution (since the response categories are ordered with respect to increasing degrees of disagreement).
Specialized methods for analyzing multinomial and ordinal multinomial response variables can be found in the Generalized Linear Models chapter.