multinomial distribution

(noun)

A generalization of the binomial distribution; gives the probability of any particular combination of numbers of successes for the various categories.

Related Terms

Examples of multinomial distribution in the following topics:

  • Categorical Data and the Multinomial Experiment

    • The multinomial experiment is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values.
    • In probability theory, the multinomial distribution is a generalization of the binomial distribution.
    • In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number $k$ of possible outcomes, with probabilities $p_1, \cdots , p_k$ (so that $p_i \geq 0$ for $i = 1, \cdots, k$ and the sum is $1$), and there are $n$ independent trials.
    • Then if the random variables Xi indicate the number of times outcome number $i$ is observed over the $n$ trials, the vector $X = (X_1, \cdots , X_k)$ follows a multinomial distribution with parameters $n$ and $p$, where $p = (p_1, \cdots , p_k)$.
    • In statistics, the multinomial experiment is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values.

  • Multinomial Distribution

    • The binomial distribution allows one to compute the probability of obtaining a given number of binary outcomes.
    • The multinomial distribution can be used to compute the probabilities in situations in which there are more than two possible outcomes.
    • The multinomial distribution can be used to answer questions such as: "If these two chess players played 12 games, what is the probability that Player A would win 7 games, Player B would win 2 games, and the remaining 3 games would be drawn?

  • Structure of the Chi-Squared Test

    • The chi-square test is used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
    • Second, we use the chi-square distribution.
    • This is a property shared by the $T$-distribution.
    • The approximation to the chi-squared distribution breaks down if expected frequencies are too low.
    • In cases where the expected value, $E$, is found to be small (indicating a small underlying population probability, and/or a small number of observations), the normal approximation of the multinomial distribution can fail.

  • t Distribution Table

    • T distribution table for 31-100, 150, 200, 300, 400, 500, and infinity.

  • Student Learning Outcomes

  • The Normal Distribution

    • Normal distributions are a family of distributions all having the same general shape.
    • The normal distribution is a continuous probability distribution, defined by the formula:
    • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
    • The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
    • Many sampling distributions based on a large $N$ can be approximated by the normal distribution even though the population distribution itself is not normal.

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
    • The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 22.
    • As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

  • The t-Distribution

    • Student's $t$-distribution (or simply the $t$-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
    • The $t$-distribution with $n − 1$ degrees of freedom is the sampling distribution of the $t$-value when the samples consist of independent identically distributed observations from a normally distributed population.
    • The $t$-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
    • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
    • Note that the $t$-distribution becomes closer to the normal distribution as $\nu$ increases.

  • Common Discrete Probability Distribution Functions

    • Your instructor will let you know if he or she wishes to cover these distributions.
    • A probability distribution function is a pattern.
    • These distributions are tools to make solving probability problems easier.
    • Each distribution has its own special characteristics.
    • Learning the characteristics enables you to distinguish among the different distributions.

  • The Standard Normal Distribution

    • The standard normal distribution is a normal distribution of standardized values called z-scores.
    • For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.
    • The mean for the standard normal distribution is 0 and the standard deviation is 1.
    • The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .
    • The value x comes from a normal distribution with mean µ and standard deviation σ.