Multinomial Process

A Multinomial Process, [math]\displaystyle{ B(n, p) }[/math], is a compound random experiment composed of [math]\displaystyle{ n }[/math] mutually independent multinomial trials with [math]\displaystyle{ p }[/math] probability mass function.

• AKA: Multinomial Experiment, Multinomial Trial Process, Multinomial Random Experiment, Compound k-Sided Coin Toss Experiment.
• Context:
  • It can range from being an Unbiased k-Sided Coin Toss Experiment to being a Biased k-Sided Coin Toss Experiment.
• Example(s):
  • a 6-Sided Dice Toss Experiment.
  • …
• Counter-Example(s):
  • a Binomial Process.
  • a Continuous Value Process.
• See: Multinomial Experiment, Multinomial Mass Function, Multivariate Normal Distribution.

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References

2009

• http://stattrek.com/Tables/multinomial.aspx#experiment
  • A multinomial experiment is a statistical experiment that has the following characteristics:
    • The experiment involves one or more trials.
    • Each trial has a discrete number of possible outcomes.
    • On any given trial, the probability that a particular outcome will occur is constant.
    • All of the trials in the experiment are independent.
  • Tossing a pair of dice is a perfect example of a multinomial experiment. Suppose we toss a pair of dice three times. Each toss represents a trial, so this experiment would have 3 trials. Each toss also has a discrete number of possible outcomes - 2 through 12. The probability of any particular outcome is constant; for example, the probability of rolling a 12 on any particular toss is always 1/36. And finally, the outcome on any toss is not affected by previous or succeeding tosses; so the trials in the experiment are independent.
  • A multinomial distribution is a probability distribution. It refers to the probabilities associated with each of the possible outcomes in a multinomial experiment. For example, suppose we toss a toss a pair of dice one time. This multinomial experiment has 11 possible outcomes: the numbers from 1 to 12. The probabilities associated with each possible outcome are an example of a multinomial distribution

• http://www.cs.bgu.ac.il/~kobbi/courses/complexity_06/Assignments/ex5.pdf
  • Exercise: Consider the following procedure for flipping k-sided coins given two-sided coins:
    • 1. Flip a two-sided coin for dlog2 ke times; the result is a binary number r.
    • 2. If r 2 [0, ..., k − 1] output r. Otherwise, repeat the process.
      • (a) Give an upper bound on the expected number of repetitions in the above procedure.
      • (b) What is the probability of no success after t repetitions?
      • (c) To flip n k-sided coins, repeat the above procedure n time. Show that the probability you will need more than 2n log k coins is exponentially small.
• http://www.k-state.edu/stats/tch/dubnicka/stat510/handout9.pdf

2006

• Suzanne R. Dubnicka. (2006). “Special Multivariate Distributions. Kansas State University, STAT 510: Handout 9
  • TERMINOLOGY : A multinomial experiment is simply a generalization of a bi-nomial experiment. In particular, consider an experiment where
    • the experiment consists of n trials (n is fixed),
    • the outcome for any trial belongs to exactly one of k ≥ 2 classes,
    • the probability that an outcome for a single trial falls into class i is given by
      • pi, for i = 1, 2, . . ., [math]\displaystyle{ k }[/math], where each pi remains constant from trial to trial (and
      • p1 + p2 + · · · + pk = 1), and
    • trials are independent.
• http://www.math.uah.edu/stat/bernoulli/Multinomial.xhtml
  • A multinomial trials process is a sequence of independent, identically distributed random variables X=(X1,X2,...) each taking k possible values. Thus, the multinomial trials process is a simple generalization of the Bernoulli trials process (which corresponds to k=2). For simplicity, we will denote the set of outcomes by {1,2,...,k}, and we will denote the common probability density functionof the trial variables by pi=ℙ(Xj=i), i∈{1,2,...,k}.

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