above and as the sum of all parameters, we can calculate the moments of the distribution. The first moment vector has coordinates:
The covariance matrix has elements:
and when
The covariance matrix is singular (its determinant is zero).
Finally, the univariate marginal distributions are all beta with parameters . All these are in the reference (see Balakrishnan and Nevzorov 2004).
Please refer to Lin (2016) for the proof of the properties of the Dirichlet distribution.
2.3.2 Multinomial Distribution
We begin with a definition of the binomial distribution.
Definition 2.24 (Binomial distribution) A random variable is said to have a binomial distribution with parameters and if it has a pmf shown below
where is the probability of success on an individual trial and is number of trials in the binomial experiment.
The multinomial distribution is a generalization of the binomial distribution. Specifically, assume that independent distributions may result in one of the outcomes generically labeled , each with corresponding probabilities . Now define a vector , where each of the counts the number of outcomes in the resulting sample of size . The joint distribution of the vector is
In the same way as the binomial probabilities appear as coefficients in the binomial expansion of , the multinomial probabilities are the coefficients in the multinomial expansion , so they sum to 1. This expansion in fact gives the name of the distribution.
If we label the outcome as a success and everything else a failure, then simply counts successes in independent trials and thus . Thus, the first moment of the random vector and the diagonal elements in the covariance matrix are easy to calculate as and , respectively. The off‐diagonal elements (covariances) are not that complicated to calculate either. However, for multinomial random vectors, the first two moments are difficult to compute. The one‐dimensional marginal distributions are binomial; however, the joint distribution of , the first components, is not multinomial. Instead, suppose we group the first categories into 1 and we let . Because the categories are linked, that is, , we also have that . We can easily verify that the vector
