Basic structure of typical multivariate data set:
Case by variables: data in matrix. Each row is a case, each column is a variable.
Example: Fisher's iris data: 5 rows of 150 by 5 matrix:
Vector valued random variable: function such that, writing ,
Cumulative Distribution Function (CDF) of : function on defined by
Defn: Distribution of rv is absolutely continuous if there is a function such that
Defn: Any satisfying () is a density of .
For most is differentiable at and
Basic tactic: specify density of
Tools: marginal densities, conditional densities, independence, transformation.
Marginalization: Simplest multivariate problem
is the marginal density of and the joint density of but they are both just densities. ``Marginal'' just to distinguish from the joint density of .
Def'n: Events and are independent if
Def'n: , are independent if
Def'n: and are independent if
Def'n: Rvs independent:
Theorem: If are independent and then are independent. Moreover, and are independent.
Conditional density of given :
Suppose with having density . Assume is a one to one (``injective") map, i.e., if and only if . Find :
Step 1: Solve for in terms of : .
Step 2: Use basic equation:
Equivalent formula inverts the matrix:
Example: The density
Solve for in terms of :
Next: marginal densities of , ?
Factor as where
Remark: easy to check .
Thus: have proved original bivariate normal density integrates to 1.
Put . Get