2 Introduction to Imprecise Probabilities
47
Definition 2.1 Let Ω be a sample space and A a σ -field (a collection of
subsets) over Ω. We will call a set function P : A → R a probability measure
if:
• ∀E ∈ A : P (E) ∈ [0, 1].
• P (Ω) = 1.
• ∀E i ∈ A, which are mutually disjoint, : P (∪ ∞
i=1 E i ) =
∞
i=1 P (E i ).
To distinguish the set function P from the ones which will follow, we will call
the tuple K := (Ω, A, P ) from Definition 2.1 a (K-)probability field.
Let us again consider the experiment with the set of possible outcomes Ω. If
the law of outcomes can be described by a probability distribution P , it indicates
that over multiple repetitions of the (exactly the same) experiment, for any chosen
E ∈ A, the relative number of outcomes which will be elements of E, will converge
to P (E) as the number of repetitions increases. The distance can be again described
probabilistically and will be mentioned again in Theorem 2.3.
We will now introduce the RVs.
Informally, consider a sample space which contains “all the possible states of
the universe”, all the possible collapses of wave functions, whimsies of Maxwell
demons, of the gods of four winds, etc. Imagine that there exists a measuring device,
a different one for each of the quantities of our interest, which is able to read the
state of the universe and return the value which our QoI has obtained at the moment.
This is how the RVs work.
Definition 2.2 Let (Ω, A, P ) be a K-probability field and (Ω X , A X ) a
measurable space. Any A X -A-measurable function X : Ω → Ω X will be
called a RV (RV).
The above definition puts emphasis on the fact that a RV is a function from the
sample space—like the measuring device from the informal definition. Now, we
are interested in how we can derive the distribution of the RVs. The probability
measure from Definition 2.1 calculates the probabilities of events in the sample
space. We need to propagate this model to assess the statements about RVs. The
answer will also allow us to specify the distributions of other derived quantities,
such as Z = f (X, Y ).
Definition 2.3 For an arbitrary mapping f : X → Y , for any set E ⊂ Y , we
define its pre-image as
f
−1 (E) := {a ∈ X : f (a) ∈ E}.
47
Definition 2.1 Let Ω be a sample space and A a σ -field (a collection of
subsets) over Ω. We will call a set function P : A → R a probability measure
if:
• ∀E ∈ A : P (E) ∈ [0, 1].
• P (Ω) = 1.
• ∀E i ∈ A, which are mutually disjoint, : P (∪ ∞
i=1 E i ) =
∞
i=1 P (E i ).
To distinguish the set function P from the ones which will follow, we will call
the tuple K := (Ω, A, P ) from Definition 2.1 a (K-)probability field.
Let us again consider the experiment with the set of possible outcomes Ω. If
the law of outcomes can be described by a probability distribution P , it indicates
that over multiple repetitions of the (exactly the same) experiment, for any chosen
E ∈ A, the relative number of outcomes which will be elements of E, will converge
to P (E) as the number of repetitions increases. The distance can be again described
probabilistically and will be mentioned again in Theorem 2.3.
We will now introduce the RVs.
Informally, consider a sample space which contains “all the possible states of
the universe”, all the possible collapses of wave functions, whimsies of Maxwell
demons, of the gods of four winds, etc. Imagine that there exists a measuring device,
a different one for each of the quantities of our interest, which is able to read the
state of the universe and return the value which our QoI has obtained at the moment.
This is how the RVs work.
Definition 2.2 Let (Ω, A, P ) be a K-probability field and (Ω X , A X ) a
measurable space. Any A X -A-measurable function X : Ω → Ω X will be
called a RV (RV).
The above definition puts emphasis on the fact that a RV is a function from the
sample space—like the measuring device from the informal definition. Now, we
are interested in how we can derive the distribution of the RVs. The probability
measure from Definition 2.1 calculates the probabilities of events in the sample
space. We need to propagate this model to assess the statements about RVs. The
answer will also allow us to specify the distributions of other derived quantities,
such as Z = f (X, Y ).
Definition 2.3 For an arbitrary mapping f : X → Y , for any set E ⊂ Y , we
define its pre-image as
f
−1 (E) := {a ∈ X : f (a) ∈ E}.