Therefore,
Definition 2.16 (Negative semidefinite matrix) A matrix
We state the following theorem without proof.
Theorem 2.1
A 2 by 2 symmetric matrix
is:
1 positive definite if and only if and det
2 negative definite if and only if and det
3 indefinite if and only if det .
2.3 Random Variables and Distribution Functions
We begin this section with the definition of
Definition 2.17 (σ‐algebra) A
1 .
2 If then its complement .
3 If is a countable collection of sets in then their union .
Definition 2.18 (Measurable functions) A real‐valued function defined on
is called measurable with respect to a sigma algebra
in that space if the inverse image of the set
, defined as
is a set in
‐algebra
, for all Borel sets
of
. Borel sets are sets that are constructed from open or closed sets by repeatedly taking countable unions, countable intersections and relative complement.
Definition 2.19 (Random vector) A random vector is any measurable function defined on the probability space
with values in
(Table 2.1).
Measurable functions will be discussed in detail in Section 20.5.
Suppose we have a random vector defined on a space
. The sigma algebra generated by
is the smallest sigma algebra in
that contains all the pre images of sets in
through
. That is
This abstract concept is necessary to make sure that we may calculate any probability related to the random variable .
Any random vector has a distribution function, defined similarly with the one‐dimensional case. Specifically, if the random vector has components
, its cumulative distribution function or cdf is defined as:
Associated with a random variable and its cdf
is another function, called the probability density function (pdf) or probability mass function (pmf). The terms pdf and pmf refer to the continuous and discrete cases of random variables, respectively.
Table 2.1 Examples of random vectors.
| Experiment | Random variable |
|---|---|
| Toss two dice |
|