Chapter 2: Random Variables and Probability Distributions
By
Joan Llull
*
Probability and Statistics.
QEM Erasmus Mundus Master. Fall 2015
Main references:
— Mood: I.3.2-5; II; III.2.1, III.2.4, III.3.1-2; V.5.1; Appendix A.2.2, A.2.4
— Lindgren: 1.2; 3.1 to 3.3, 3.5; 4.1 to 4.6, 4.9, 4.10; 6.1, 6.5, 6.8; (3.2)
I.
Preliminaries: An Introduction to Set Theory
We start this chapter with the introduction of some tools that we are going to
use throughout this course (and you will use in subsequent courses).
First, we
introduce some definitions, and then describe some operators and properties of
these operators.
Consider a collection of objects, including all objects under consideration in a
given discussion. Each object in our collection is an
element
or a point. The
totality of all these elements is called the
space
, also known as the universe, or
the universal set, and is denoted by Ω. We denote an element of the set Ω by
ω
.
For example, a set can be all the citizens of a country, or all the points in a plane
(i.e. Ω =
R
2
, and
ω
= (
x, y
) for any pair of real numbers
x
and
y
). A partition of
the space Ω is called a
set
, which we denote by calligraphic capital Latin letters,
with or without subscripts. When we opt for the second, we define the catalog of
all possible incides as the
index set
, which we denote by Λ (for example, if we
consider the sets
A
1
,
A
2
, and
A
3
, then Λ =
{
1
,
2
,
3
}
.
To express that an element
ω
is part of a set
A
, we write
w
∈ A
, and to state
the opposite, we write
w /
∈ A
. We can define sets by explicitly specifying all its
elements (e.g.
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
), or implicitly, by specifying properties that
describe its elements (e.g.
A
=
{
(
x, y
) :
x
∈
R
, y
∈
R
+
}
). The set that includes
no elements is called the
empty set
, and is denoted by
∅
.
Now we define a list of operators for sets:
•
Subset
: when all elements of a set
A
are also elements of a set
B
we say
that
A
is a subset of
B
, denoted by
A ⊂ B
(“
A
is contained in
B
”) or
B ⊃ A
*
Departament d’Economia i Hist`
oria Econ`
omica.
Universitat Aut`
onoma de Barcelona.
Facultat d’Economia, Edifici B, Campus de Bellaterra, 08193, Cerdanyola del Vall`
es, Barcelona
(Spain). E-mail: joan.llull[at]movebarcelona[dot]eu. URL: .
1

(“
B
contains
A
”).
•
Equivalent set:
two sets
A
and
B
are equivalent or equal, denoted
A
=
B
if
A ⊂ B
and
B ⊂ A
.
•
Union
: the set that consists of all points that are either in
A
, in
B
, or in
both
A
and
B
is defined to be the union between
A
and
B
, and is denoted
by
A ∪ B
. More generally, let Λ be an index set, and
{A
λ
} ≡ {
A
λ
:
λ
∈
Λ
}
,
a collection of subsets of Ω indexed by Λ. The set that consists of all points
that belong to
A
λ
for at least one
λ
∈
Λ is called the union of the sets
{A
λ
}
,
denoted by
∪
λ
∈
Λ
A
λ
. If Λ =
∅
, we define
∪
λ
∈
∅
A
λ
≡
∅
.

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