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Since the variance is always nonnegative, the covariance matrix must be nonnegative definite (or positive semidefinite). We recall that a square symmetric matrix is positive semidefinite if . This difference is in fact important in the context of random variables since you may be able to construct a linear combination which is not always constant but whose variance is equal to zero.
The covariance matrix is discussed in detail in Chapter 3.
We now present examples of multivariate distributions.
2.3.1 The Dirichlet Distribution
Before we discuss the Dirichlet distribution, we define the Beta distribution.
Definition 2.22 (Beta distribution) A random variable is said to have a Beta distribution with parameters and if it has a pdf defined as:
where and .
The Dirichlet distribution , named after Johann Peter Gustav Lejeune Dirichlet (1805–1859), is a multivariate distribution parameterized by a vector of positive parameters .
Specifically, the joint density of an ‐dimensional random vector is defined as:
where is an indicator function.
Definition 2.23 (Indicator function) The indicator function of a subset of a set is a function
defined as
The components of the random vector thus are always positive and have the property . The normalizing constant is the multinomial beta function, that is defined as:
where we used the notation and for the Gamma function.
Because the Dirichlet distribution creates positive numbers that always sum to 1, it is extremely useful to create candidates for probabilities of possible outcomes. This distribution is very popular and related to the multinomial distribution which needs numbers summing to 1 to model the probabilities in the distribution. The multinomial distribution is defined in Section 2.3.2.
With the notation
