We begin by discussing the measurability of set-valued maps and then introduce the definition of an interval random variable. The basic definitions and more details can be found in . A measurable space consists of a basic set Ω together with a σ-algebra of subsets of Ω called measurable sets. Here, we consider closed convex value set-valued maps , i.e., is a closed convex subset of for each . This is the case when F is interval valued. The latter notion means that for each , the components of are closed intervals in ℝ.
We first define what it means for a set-valued map to be measurable. Recall that the inverse image of a set under the set-valued map F is defined by
and that the graph of F (denoted by ) is defined by
Definition 3 Let be a measurable space and be a set-valued map. F is called measurable if the inverse image of each open set is a measurable set: if is open, then .
We are now in a position to introduce the definition of interval random variables and interval stochastic processes.
Definition 4 Let be a probability space. An interval-valued map is called an interval random variable if
X is measurable, and
the function is continuous on X, where is the probability density function for the random variable x.
An interval stochastic process is an indexed set of interval random variables.
The probability density function is then the interval-valued function
In order to study the expectations and variances of interval random variables, we need to discuss first the integral of set-valued maps and, in particular, interval-valued maps. The discussion begins with the notion of measurable selections.
Definition 5 Let be a measurable space and be a measurable set-valued map. A measurable selection of F is a measurable map satisfying for each .
It is well known that every measurable set-valued map has at least one measurable selection . Furthermore, we have the following equivalences .
Theorem 6 Let be a measurable space and denote by ℬ the σ-algebra of Borel sets in . Let be a set-valued map. The following are equivalent.
F is measurable.
for every .
There exists a sequence of measurable selections
for each .
A countable family of measurable selections satisfying the last property is called dense.
Let be an interval-valued map. We define the two special functions and such that and , where for each . The next lemma shows that and are measurable selections of F when the latter is measurable.
Lemma 7 Let be a measurable interval-valued map. Then the point functions and are measurable selections of F.
Proof Choose a sequence of measurable selections of F such that
Then and (here the inf and sup operations are taken componentwise). Since the inf and the sup operators preserve measurability, we see that the functions and are measurable selections of F. □
Example Let and define by
Let be an enumeration of the rational numbers in the interval , and let us assume that , . Define by
Thus, and . For every , the set is dense in the interval . Thus, .
Now suppose that is a measure space and is a set-valued map. A measurable selection f of F is an integrable selection if f is integrable with respect to the measure μ. The set of all integrable selections of F will be denoted by ℱ. The map F is called integrably bounded if there exists a μ-integrable function such that for μ-almost every . Here, B denotes the unit ball in . In this case, every measurable selection f of F is also an integrable selection since implies that , where denotes the Euclidean norm on .
Definition 8 The integral of a set-valued map F is defined to be the set of integrals of integrable selections of F. That is,
We shall say that F is integrable if every measurable selection is integrable.
We have the following immediate properties:
Lemma 9 Let be an interval-valued map. If and are integrable, then F is integrable and
Proof The first equality is shown as follows. Since for every and every integrable selection f of F we have ,
On the other hand, let . We may write for some . Then
where . Hence, .
The second equality is an immediate consequence of this. □
It will always be assumed that both and are integrable.
Example Let Ω and F be defined as in the previous example. Let μ be the measure defined by
In view of (3), we have the following corollary.
Corollary 10 Let be integrable interval-valued maps. Then
Let be a probability space, and let be an interval random variable. We have
We shall say that Z is normally distributed if each is normally distributed. An interval stochastic process will be called normally distributed if for each , is normally distributed.
Let Z be an interval random variable. Then for each ,
By the continuity of ,
This means that
Guided by this and Lemma 9, we can define the interval expectation of the interval random variable Z as follows.
Definition 11 The interval expectation of an interval random variable Z is defined as
This definition coincides with Definition 2 since
It should also be noted that the expectation of a vector random variable is the vector of expectations of its components.
It follows from equations (2) and (3) that
Also, if and Z is an interval random variable, then
The same is true if I is an interval vector and Z is an interval random variable.
More generally, if A is a interval matrix and if its columns are denoted by the interval vectors , then
To introduce covariance of two interval random variables Y, Z, we need to assume that the function is continuous on . Here, is the joint probability density function of the two random variables x, y.
Definition 12 The interval covariance of two interval random variables Y, Z is defined as
To see that is an interval, note that
If , we get the definition of the variance of an interval random variable Z as
which is also an interval. Elementary calculus considerations reveal that
where , , . This last equation provides a formula for computing the interval .
For interval random vectors, the above definitions hold componentwise.
The two interval random variables Y, Z will be called uncorrelated if for each , , , are uncorrelated. Therefore, Y, Z are uncorrelated if and only if .
It is now straightforward to check the following theorem.
Theorem 13 Let , , be interval random vectors, and let , , , then
The assumed continuous dependence of the probability density function (joint density function) on the random variable (variables) in an interval random variable (interval random variables) implies that the conditional probability density function is also continuous. This guarantees that the generalization of the conditional density function to the interval setting is always an interval.
Definition 14 The interval conditional expectation is defined as
The following theorem is easily checked.
Theorem 15 For vector random variables X, Y, Z and interval matrix A of appropriate dimensions,