Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 117

Appendix
Basic Notions of Probability Theory

Abstract In this Appendix, the basic concepts of Probability Theory employed in
the topic are explained and their main properties are discussed.

Keywords Probability, Expected value, Additivity, Variance, Covariance,
Independence, Conditional distribution

A.1 The Concept of Probability

A probability distribution, or, succinctly, a probability, on a finite set S, is an

additive function p, from the set of parts of S, P(S), to the interval [0, 1], satisfying
p(S) = 1. Additivity in a function whose arguments are sets means that, for any pair
of disjoint sets A 1 and A 2 ,
p ð A 1 [ A 2 Þ ¼ pA ðÞþ pA ð :

In a reference to statistics practice, the sets that enter as arguments of a
probability function are called events, the elements of S are called outcomes and S
is called a sample space.
This concept may be extended from finite sets to any set S, but the set of events

must present certain properties that sometimes may not be presented by the set P(S) of
all parts of S. In the general de

nition, the set of events may be any nonempty subset
Λ (S) of P(S) closed for complements and countable unions (a set of events satisfying
these probabilities is called a sigma algebra). More precisely, a sigma algebra is any
subset

Λ (S) of P(S) satisfying: (1) if A

∈ Λ (S) then S\A ∈ Λ (S) and (2) if A i ∈ Λ (S) for

every set A i of the sequence of sets {A i } i ∈ N , then ∪ i ∈ N A i ∈ Λ (S).

A function p with domain

Λ (S) and satisfying p(S) = 1 is then a probability in

(S,

Λ (S)) if and only if, for every sequence {A i } i ∈ N of elements of Λ (S) satisfying

A i ∩ A j = Φ if i ≠ j,

X

p ð[ i 2 N A i Þ ¼ pA ð :

i 2 N