Multinomial Distribution

Multinomial distribution is a multivariate version of the binomial distribution. It is the probability distribution of the outcomes from a multinomial experiment. It is used in the case of an experiment that has a possibility of resulting in more than two possible outcomes.

The simplest technique to construct a multinomial random variable is to replicate an experiment (by drawing n uniform random numbers and assigning them to certain bins based on the cumulative value of the p vector) to produce a multinomial random variable. In instructional statistics, this distribution is put to various uses.

• The multinomial distribution represents the likelihood of receiving a certain set of counts where each trial has a discrete number of possible outcomes.
• The most direct goodness-of-fit test is based on the multinomial distribution of response patterns.
• Its applications and use cases frequently involve the evaluation of the likelihood of a set of outcomes that are usually more than two or at least two.
• It is a type of probability distribution of which binomial distribution is a subtype.

The multinomial distribution is a multivariate discrete distribution. Multinomial experiments include the following characteristics:

• The experiment comprises of n repeated trials.
• Its results have statistical significance.
• There are a finite number of possible outcomes for each trial, and the likelihood of any event occurring is constant throughout the experiment.
• The results of one experiment do not influence the results of the others.

Assuming the model is valid, the most straightforward way to determine a model's fit is to use the multinomial distribution of response patterns. Accordingly, there are 'nm' potential response patterns ranging from (l,...,l) to (m,...,m) for n items with m response categories for each item. Hence, all response patterns have the same probability of occurring if u is a typical pattern.

Thus, to estimate the frequencies of the response patterns, use multinomial distribution with parameters n and actual probability for all the response patterns. According to the multivariate central limit theorem, the multivariate normal distribution can approximate the distribution for large sample sizes.

While considering the entire data, the distribution of the observations has a multinomial shape for observations from different Poisson distributions. Probabilities in the multinomial distribution are based on the Poisson mean for each cell multiplied by all Poisson mean values.

A multinomial experiment has a subtype known as a binomial one. In this regard, there is one major distinction. Accordingly, in a binomial experiment, there are only two possibilities for each trial. At the same time, each experiment in a multinomial trial has the potential difference for two or more different results.

For instance, you perform n times an experiment with K outcomes. Then, you denote by Xi the number of times you obtain the i-th outcome. In that case, the random vector X is defined as  X = is a multinomial random vector.

The multinomial distribution is the generalization of the binomial distribution to the case of n repeated trials where there are more than two possible outcomes for each. If an event may occur with k possible outcomes, each with a probability, pi(i = 1,1,…,k), with ∑^k_(i=1)pi = 1, and if r_i is the number of the outcome associated with 'pi' occurs, then the random variables r_i (i = 1,2,…,k-1) have a multinomial probability defined as

f (r₁,r₂,....,r_k-1) = n! ∏^k_i=1 pi^ri/∏^k_i=1 r_i!, r_i = 0,1,2,....,n.

Note that each of the r_i ranges from 0 to n inclusive with only (k-1) variables because of the linear constraint:

∑^k_i=1 r_i = n

Just as the binomial distribution tends to the univariate normal, so does the multinomial distribution tend to limit to the multivariate normal distribution.

Suppose a random variable X has a multinomial distribution. In that case, the following multinomial distribution calculator calculates the likelihood that event 1 occurs exactly x1 times, event 2 occurs exactly x2 times, event 3 occurs exactly x3 times, and so on. Hence following is the multinomial distribution formula:

Probability = n!*(p1x1 * p2x2 * ... * pkxk)/(x1!*x2!*...*xk!)

Where:

• n: the total number of events
• x1, x2, xk: the number of occurrences of event 1, event 2, and event k, respectively.
• p1, p2, pk: the likelihood that results in 1, 2, and k happen, respectively, in a trail.

Let us have a look at the multinomial distribution example to understand the concept better:

Rebecca, a portfolio manager, utilizes it to assess the probability of her client's investment. For 60% of the time, she chooses a small-cap index to outperform a large-cap index. Whereas for 40% of the time, Rebecca opts for a large-cap index to outperform a small-cap index. For 10% of the time, the indexes may have the same or approximate return.

Since the trial may last a full year of trading days in such cases, Rebecca uses actual market data to validate the outcomes. In scenarios where the likelihood of this set of occurrences is high, her clients may tempt to overinvest in the small-cap index.

Likewise, Neil, a financial analyst, uses this method to evaluate the likelihood of events, like potential quarterly sales for a business when its competitors post lower-than-expected profits.

This article is a guide to Multinomial Distribution & its definition. We explain its properties, formula, calculator, comparison with binomial, & example. You can learn more from the following articles -

• Probability Distribution
• Regression
• Negatively Skewed Distribution