Multinomial distribution
Introduction
The Multinomial distribution is a generalization of the binomial distribution. It describes the outcomes of multi-nomial scenarios unlike binomial where scenarios must be only of two kinds. In other words, it models the probability of counts for rolling a k-sided die n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
Definition
Let X be a random variable that counts the number of outcomes of each category from a multinomial experiment. Then X is said to have a multinomial distribution, denoted by X ~ Mult(n, p), where n is the number of trials and p is the vector of success probabilities for the k categories.
The probability mass function of a multinomial distribution is given by:
P(X1 = x1, X2 = x2, ..., Xk = xk) = n! / (x1!x2!...xk!) * (p1^x1 * p2^x2 * ... * pk^xk)
where x1, x2, ..., xk are nonnegative integer outcomes such that x1 + x2 + ... + xk = n, and p1, p2, ..., pk are the probabilities of the k categories such that p1 + p2 + ... + pk = 1.
Properties
1. The expected value of a multinomially distributed random variable X is E[X] = np, where n is the number of trials and p is the vector of success probabilities.
2. The variance of a multinomially distributed random variable X is Var[X] = np(1 - p) for the diagonal elements and -npipj for the off-diagonal elements, where pi is the success probability for the i-th category.
3. The multinomial distribution is a multivariate generalization of the binomial distribution. For k = 2, the multinomial distribution reduces to the binomial distribution.
4. The multinomial distribution is the distribution of the frequencies of events in a sequence of independent categorical trials.
5. The multinomial distribution is symmetric when all pi are equal.
Applications
The multinomial distribution has wide applications in various fields such as statistics, machine learning, genetics, and marketing research. It is used to model the probabilities of different outcomes in an experiment where each trial has multiple possible outcomes. For example, in genetics, it is used to model the distribution of genetic traits in a population. In machine learning, it is used in multiclass classification problems where an object can belong to one of multiple classes. In marketing research, it is used to model the choice behavior of consumers among multiple product categories.