Multivariate Distributions

Multinomial Distribution

Definition

The multivariate version of a Binomial distribution is called a Multinomial Distribution. Consider drawing a ball from an urn which has balls with different colors. Let , where and , and suppose that is the probability of drawing a ball of color . Draw times (independent draws with replacement) and let where is the number of times that color appears. Hence, .

We say that has a distribution, written . The probability function is

where

Lemma

Suppose that , where and . The marginal distribution of is .

Multivariate Normal Distribution

Definition

A vector has a Multivariate Normal Distribution, written , if it has density

where is the determinant of , is a vector of length and is a symmetric, positive definite matrix.

Since is symmetric and positive definite, it can be shown that there exists a matrix (called the square root of ), with the following properties

• is symmetric

Theorem

If and , then . Conversely, if , then .

Suppose we partition a random Normal vector as . We can similarly partition and

Theorem

Let . Then

1. The marginal distribution of is
2. The conditional distribution of given is

3. If is a vector, then

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.