left-parenthesis 1 minus x right-parenthesis Superscript beta minus 1 Baseline comma 2nd Column if 0 less-than x less-than 1 comma 2nd Row 1st Column 0 comma 2nd Column if otherwise comma EndMatrix"/>
where
The Dirichlet distribution
Specifically, the joint density of an
where
Definition 2.23 (Indicator function) The indicator function of a subset
defined as
The components of the random vector
where we used the notation
Because the Dirichlet distribution creates
With the notation mentioned above and
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
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
