2 Introduction to Imprecise Probabilities
49
Definition 2.4 Let X be a RV on (Ω, A, P ). Then,
E P X [X] :=
Ω X
xdP X =
Ω
X(ω)dP
will be called the expected value of RV X. The subscript representing the
underlying distribution (P X ) is usually omitted, and we will also do so for
precise RVs. We introduce it due to the necessity of computing expected
values for various probability measures later in the chapter.
The expected value represents a typical value, the limit of average values of
multiple draws. By the law of large numbers, the average of the finite amount of
draws from X will converge towards E [X].
Theorem 2.2 (Law of Large Numbers) Let X 1 , . . . , X n be a series of RVs,
such that each of them is distributed according to the same law with a finite
expected value E [X 1 ] =: μ ∈ R. Then,
lim
n→∞
1
n
n
i=1
X i = μ,
where the real number μ can be viewed as a degenerate RV M s.t. ∀ω ∈ Ω :
M(ω) = μ.
The law of large number is the core principle which allows us to perform
statistical inference. It guarantees that we will approach correct assessments about
the sampling distributions (means, other moments, probability statements, etc.) with
increasing number of observations.
Another important result of the probability theory, the central limit theorem,
states what is the asymptotic convergence rate towards these values. It also provides
theoretical guarantee for convergence of Monte Carlo algorithms [28].
Theorem 2.3 (Central Limit Theorem) Let X 1 , . . . , X n be a series of RVs,
such that each of them is distributed according to the same law with a finite
expected value E [X 1 ] =: μ ∈ R and a finite variance E
(X 1 − μ) 2 =:
σ 2 ∈ R + . Then
(continued)
49
Definition 2.4 Let X be a RV on (Ω, A, P ). Then,
E P X [X] :=
Ω X
xdP X =
Ω
X(ω)dP
will be called the expected value of RV X. The subscript representing the
underlying distribution (P X ) is usually omitted, and we will also do so for
precise RVs. We introduce it due to the necessity of computing expected
values for various probability measures later in the chapter.
The expected value represents a typical value, the limit of average values of
multiple draws. By the law of large numbers, the average of the finite amount of
draws from X will converge towards E [X].
Theorem 2.2 (Law of Large Numbers) Let X 1 , . . . , X n be a series of RVs,
such that each of them is distributed according to the same law with a finite
expected value E [X 1 ] =: μ ∈ R. Then,
lim
n→∞
1
n
n
i=1
X i = μ,
where the real number μ can be viewed as a degenerate RV M s.t. ∀ω ∈ Ω :
M(ω) = μ.
The law of large number is the core principle which allows us to perform
statistical inference. It guarantees that we will approach correct assessments about
the sampling distributions (means, other moments, probability statements, etc.) with
increasing number of observations.
Another important result of the probability theory, the central limit theorem,
states what is the asymptotic convergence rate towards these values. It also provides
theoretical guarantee for convergence of Monte Carlo algorithms [28].
Theorem 2.3 (Central Limit Theorem) Let X 1 , . . . , X n be a series of RVs,
such that each of them is distributed according to the same law with a finite
expected value E [X 1 ] =: μ ∈ R and a finite variance E
(X 1 − μ) 2 =:
σ 2 ∈ R + . Then
(continued)