Maria Cristina Mariani - Data Science in Theory and Practice

Therefore, is negative definite.

Definition 2.16 (Negative semidefinite matrix)A matrix is called negative semidefinite if, for any vector , we have

We state the following theorem without proof.

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 .

We begin this section with the definition of ‐algebra.

Definition 2.17 (σ‐algebra)A ‐algebra is a collection of sets of satisfying the following condition:

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	= sum of the numbers
Toss a coin 10 times	= sum of tails in 10 tosses

Definition 2.20 (Probability mass function)The pmf of a discrete random variable is given by