50
D. Krpelík and T. Basu
Theorem 2.3 (continued)
lim
n→∞
1
√
nσ
n
i=1
(X i − μ) ∼ N (0, 1),
where N (0, 1) is the standard normal distribution.
2.3.2 Probability via Expectation
Another approach for building an axiomatic theory of probability is explained by
Whittle [44]. The idea is that, instead of focusing on the probability distributions as
measures on a sample space, we investigate the RVs from the functional perspective.
Whittle therefore describes a system based on axioms posed on the expected
values, instead of that based on the probability measures. He further shows how
to reproduce results of the standard approach (measure-based) to probability theory.
Definition 2.5 Let (Ω, A) be a measurable space and L(Ω) the set of all RVs
on Ω. We will call a functional E : L(Ω) → R the expected value if
• ∀X ∈ L(Ω) : X ≥ 0 ⇒ E(X) ≥ 0.
• ∀a, b ∈ R, ∀X, Y ∈ L(Ω) : E(aX + bY ) = aE(X) + bE(Y ).
• E(1) = 1.
• if, ∀ω ∈ Ω, a sequence X n (ω) increases monotonically to X(ω), then
E(X) = lim E(X n ).
Starting from the axioms of the expected value functional leads to the same
theoretical system as Kolmogorov’s approach. In the precise probability case,
both the approaches are equivalent, but this is no longer the case with imprecise
probabilities, where the theory of lower previsions (Sect. 2.5) allows us to represent
a larger set of models than the IP extension of the measure-theoretic approach
(Sect. 2.3).
A similar approach for formulating probability theory was also explored by
de Finetti and Savage [15, 35] from a decision-making perspective. A notable
difference introduced by de Finetti is that the expected values are called previsions
but denote the same object. Also, in the notation, de Finetti further does not
distinguish between the symbols for probability of an event, and for an expectation
of a RV, both are P , as there exists a one-to-one mapping between events and binary
RVs. We can obtain the probability of any event E ⊂ Ω by simply calculating the
expected value of its indicator function I E ∈ L(Ω). The meaning is usually evident
from the context.
D. Krpelík and T. Basu
Theorem 2.3 (continued)
lim
n→∞
1
√
nσ
n
i=1
(X i − μ) ∼ N (0, 1),
where N (0, 1) is the standard normal distribution.
2.3.2 Probability via Expectation
Another approach for building an axiomatic theory of probability is explained by
Whittle [44]. The idea is that, instead of focusing on the probability distributions as
measures on a sample space, we investigate the RVs from the functional perspective.
Whittle therefore describes a system based on axioms posed on the expected
values, instead of that based on the probability measures. He further shows how
to reproduce results of the standard approach (measure-based) to probability theory.
Definition 2.5 Let (Ω, A) be a measurable space and L(Ω) the set of all RVs
on Ω. We will call a functional E : L(Ω) → R the expected value if
• ∀X ∈ L(Ω) : X ≥ 0 ⇒ E(X) ≥ 0.
• ∀a, b ∈ R, ∀X, Y ∈ L(Ω) : E(aX + bY ) = aE(X) + bE(Y ).
• E(1) = 1.
• if, ∀ω ∈ Ω, a sequence X n (ω) increases monotonically to X(ω), then
E(X) = lim E(X n ).
Starting from the axioms of the expected value functional leads to the same
theoretical system as Kolmogorov’s approach. In the precise probability case,
both the approaches are equivalent, but this is no longer the case with imprecise
probabilities, where the theory of lower previsions (Sect. 2.5) allows us to represent
a larger set of models than the IP extension of the measure-theoretic approach
(Sect. 2.3).
A similar approach for formulating probability theory was also explored by
de Finetti and Savage [15, 35] from a decision-making perspective. A notable
difference introduced by de Finetti is that the expected values are called previsions
but denote the same object. Also, in the notation, de Finetti further does not
distinguish between the symbols for probability of an event, and for an expectation
of a RV, both are P , as there exists a one-to-one mapping between events and binary
RVs. We can obtain the probability of any event E ⊂ Ω by simply calculating the
expected value of its indicator function I E ∈ L(Ω). The meaning is usually evident
from the context.