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
- The marginal distribution of is
- The conditional distribution of given is
- If is a vector, then