multinomial distribution as 2D (triangular) slices of Pascal's pyramid, or 3D/4D/+ (pyramid-shaped) slices of higher-dimensional analogs of Pascal's triangle. This reveals an interpretation of the range of the distribution: discretized equilateral "pyramids" in arbitrary dimension—i.e. a simplex with a grid. As polynomial coefficients Similarly, just like one can interpret the binomial distribution as the polynomial coefficients of when expanded, one can interpret the multinomial distribution as the coefficients of when expanded, noting that just the coefficients must sum up to 1. Related distributions In some fields such as natural language processing, categorical and multinomial distributions are synonymous and it is common to speak of a multinomial distribution when a categorical distribution is actually meant. This stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single trial. When k = 2, the multinomial distribution is the binomial distribution. Categorical distribution, the distribution of each trial; for k = 2, this is the Bernoulli distribution. The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics. Dirichlet-multinomial distribution. Beta-binomial distribution. Negative multinomial distribution Hardy–Weinberg principle (it is a trinomial distribution with probabilities ) Statistical inference Equivalence tests for multinomial distributions The goal of equivalence testing is to establish the agreement between a theoretical multinomial distribution and observed counting frequencies. The theoretical distribution may be a fully specified multinomial distribution or a parametric family of multinomial distributions. Let denote a theoretical multinomial distribution and let be a true underlying distribution. The distributions and are considered equivalent if for a distance and a tolerance parameter . The equivalence test problem is versus . The true underlying distribution is unknown. Instead, the counting frequencies are observed, where is a sample size. An equivalence test uses to reject . If can be rejected then the equivalence between and is shown at a given significance level. The equivalence test for Euclidean distance can be found in text book of Wellek (2010). The equivalence test for the total variation distance is developed in Ostrovski (2017). The exact equivalence test for the specific cumulative distance is proposed in Frey (2009). The distance between the true underlying distribution