Multinomial Distribution


Also found in: Medical, Wikipedia.

multinomial distribution

[¦məl·tə¦nō·mē·əl ‚di·strə′byü·shən]
(mathematics)
The joint distribution of the set of random variables which are the number of occurrences of the possible outcomes in a sequence of multinomial trials.

Multinomial Distribution

 

a joint probability distribution of random variables, each of which expresses the number of times one of several mutually exclusive events occurs in repeated independent trials. Suppose that in each trial the events A1, …, Am have probabilities p1, …, pm, respectively, where 0 ≤ Pk < 1; k = 1, …, m; and p1 + ··· + pm = 1. Then the joint distribution of the random variables X1, …, xm, where Xk is the number of occurrences of the event Ak in n trials, is given by the probabilities

that is, the probability that the event A1 occurs n1 times in n independent trials, that the event A2 occurs n2 times, and so on. These probabilities are defined for every set of nonnegative integers n1, …, nm that satisfy the single condition n1 + ··· + nm = n.

The multinomial distribution is a natural generalization of the binomial distribution, to which it reduces when m = 2. Every random variable Xk in this case must have a binomial distribution with mathematical expectation npk and variance npk(1 – Pk). As n → ∞, the joint distribution of the random variables

tends to some limiting normal distribution, and the sum

approaches a chi-square distribution with n – 1 degrees of freedom. This sum is used in mathematical statistics in the chi-square test.

REFERENCES

Cramer, H. Matematicheskie metody statistiki. Moscow, 1948. (Translated from English.)
Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, 2nd ed., vols. 1–2. Moscow, 1967. (Translated from English.)

A. V. PROKHOROV

References in periodicals archive ?
To estimate the expected frequencies and the model parameters, the estimated and observed values were compared with a multinomial distribution expressed as a negative log-likelihood (-ln [L.sub.2]) (Haddon 2001):
The aim of the current work is to model hierarchical data from the components of the seed germination test of Stevia rebaudiana using the binomial, beta-binomial and multinomial distributions to discriminate effects from fixed and random explanatory variables.
where the document is collected by learner l, z' is a topic, K is all topics, [theta]' is the multinomial distribution of the learner l over topics, [0'.sub.z], is the multinomial distribution of the topic z' over words.
(3.2)Generate a multinomial distribution over non-visual topics:[[theta].sub.d]~Dir([[alpha].sup.w.sub.0])
Observation error in survey age-composition data from the operating model was generated with the multinomial distribution. To evaluate the influence of age-composition sample size in a fishery-independent survey on SCAA model results, 13 sample sizes were used to generate trawl survey age composition data that ranged from 10 to 100,000 (by multiples of 2.5 and 2, e.g., 10, 25, 50, 100, 250, 500 ...).
(b) Generate function f from multinomial distribution Mult([theta]), that is, f ~ Mult([theta]).
In this study, because the assumption of constant volatility has its defects and the assumption of stochastic volatility is complex to calculate, we assume that the annualized volatility of the price of mineral products follows a multinomial distribution. In order to take into consideration the notion that the annualized volatility of the price of mineral products may experience relatively significant variations from year to year, we propose a combined multinomial pricing model for pricing mining concessions.
where rank(k,v) is the ranking of clinical activity v in topic k based on the multinomial distribution [[phi].sub.k].
Then we sample from the multinomial distribution based on p to pick a location I within the region.
Let each of the N observations on the dependent variable [Y.sub.i] (i = 1, ..., N) is treated as a single draw from a multinomial distribution with M outcomes and define a dummy variable [[delta].sub.ij] = 1 if person / makes choice j, [[delta].sub.ij] = 0, otherwise (j = 1, ..., N), then the likelihood of this sample will be
For the purpose of analysing the distribution of the effort along the project lifecycle the data set under study can be viewed as a contingency table whose observations follow the multinomial distribution M([p.sub.r], [N.sub.r]), where [N.sub.r] is known for each [p.sub.r].
Symbol Description K Number of topics M Number of documents V Number of unique words A Number of unique authors [N.sub.m] Number of word tokens in document m [A.sub.m] Number of authors in document m a Single author index, a [member of] [1, A] k Single topic index, k [member of] [1, K] m Single document index, me [1, M] n Single word token index, m [member of] [1, [N.sub.m]] V Single word index, v [member of] [1, V] [a.sub.m] Authors in document m, [a.sub.m] [subset or equal to] [1, A] [[??].sub.a] Multinomial distribution of topics specific to the author a.