given extraction will be in color i. The probability mass function of this multinomial distribution is: for non-negative integers x1, ..., xk. The probability mass function can be expressed using the gamma function as: This form shows its resemblance to the Dirichlet distribution, which is its conjugate prior. Example Suppose that in a three-way election for a large country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes. If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters for candidate B and three supporters for candidate C in the sample? Note: Since we’re assuming that the voting population is large, it is reasonable and permissible to think of the probabilities as unchanging once a voter is selected for the sample. Technically speaking this is sampling without replacement, so the correct distribution is the multivariate hypergeometric distribution, but the distributions converge as the population grows large in comparison to a fixed sample size. Properties Expected value and variance The expected number of times the outcome i was observed over n trials is The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore The off-diagonal entries are the covariances: for i, j distinct. All covariances are negative because for fixed n, an increase in one component of a multinomial vector requires a decrease in another component. When these expressions are combined into a matrix with i, j element the result is a k × k positive-semidefinite covariance matrix of rank k − 1. In the special case where k = n and where the pi are all equal, the covariance matrix is the centering matrix. The entries of the corresponding correlation matrix are Note that the sample size drops out of this expression. Each of the k components separately has a binomial distribution with parameters n and pi, for the appropriate value of the subscript i. The support of the multinomial distribution is the set Its number of elements is Matrix notation In matrix notation, and with = the row vector transpose of the column vector . Visualization As slices of generalized Pascal's triangle Just like one can interpret the binomial distribution as (normalized) one-dimensional (1D) slices of Pascal's triangle, so too can one interpret the