     # Interval probability density functions constructed from a generalization of the Moore and Yang integral

Moore and Yang defined an integral notion, based on an extension of Riemann sums, for inclusion monotonic continuous interval functions, where the integrations limits are real numbers. This integral notion extend the usual integration of real functions based on Riemann sums. In this paper, we extend this approach by considering intervals as integration limits instead of real numbers and we abolish the inclusion monotonicity restriction of the interval functions and this notion is used to determine interval probability density functions.

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## 1 Introduction

The classical probability theory provides a mathematical model for the study of uncertainties of a random nature, also called uncertain knowledge that, even if produced identically, presents to each experiment, results that vary unpredictably. In it, random events must be precisely defined, which is often not the case in real situations. Therefore, in the classical theory of probabilities consider an absolute knowledge without consider the uncertainties generated from incomplete or imprecise knowledge present in many real situations and to deal with these uncertainties, were proposal several ways to extend the notion of probabilities

[2, 13, 20, 27, 46, 49]. Within this context, many researchers have developed studies on inaccurate probabilities, in order to consider problems that are not contemplated by classical theory [23, 29]. Among them, the interval mathematics has served as a theoretical framework for this purpose as one can see on the works Yager , Sarveswaram et al. , Tanaka and Sugihara , Intan , Zhang, et al. , Campos and dos Santos , and Jamison and Lodwick .

In this work we are interested in studying the interval case of the probability density functions constructed from integrals. Also known as density of a continuous random variable, these functions describe the probability of a random variable assuming that the value of the random variable would equal that sample the given value in the sample space, which can be calculated from the integral of the density of that variable in a given range, i.e. given a continuous variable

and the values set , the probability density function asigns to each element a number satisfying properties: (1) and (2) . In this framework some researchers have been studied the interval version of probability density functions in order to provide a model where the uncertainty of the varariable is measured by an interval. For instance, Berleant in  has developed an system to evaluate the reasoning automatically by using an intervals probability density functions. Ramírez 

used a data sampling interval to analyze its influence in the estimation of the parameters of the Weibull wind speed probability density distribution. In both cases authors has been used interval to estimate the uncertainty considering the usual version of probability density function, i.e. they not consider an interval version of probability density function indeed. Here, we propose an interval probability density interpretation based on a new way to define interval integrals.

In  Moore and Yang defined the first approach to interval integrals. In  further properties of this interval integral notions and their relation with integration of ordinary real functions are established. The Moore-Yang integration of interval functions approach is based on a generalization of Riemann sums for real integrals. Moore in  defined integrals of continuous function from into which can be seen as restrictions of continuous and monotonic interval functions to degenerate intervals (this is guarantee by the dependency on the degenerate intervals of interval integral ).

There exist some other few interval integration approaches in the literature. Among other, are the work of Caprani, Madsen and Rall in , they defined integrals for functions (not necessarily continuous) from an interval to the set of interval of extended real numbers (i.e. real numbers with and ) based on Darboux integrals. The Caprani, Madsen and Rall approach have Moore integrals (as defined in ) as an special case. Lately, Rall in , considering this interval integration approach, partially solved the problem of assignment of infinite intervals to some improper intervals known to be finite. Corliss in  extend the Caprani, Madsen and Rall notion of integral for considering interval valued limits. More recently, we can cite the following works on integration of interval-valued functions [5, 10, 24].

The Escardó approach in  adapted the Edalat work , who extended the Riemann integrals using Scott domain theory, for the domain of intervals, as seen in , resulting in a new version of the Moore-Yang integrals, which consider Scott continuous functions instead of continuous (w.r.to Moore metrics) and inclusion monotonic functions (notice that all Scott continuous interval functions are inclusion monotonic). The relationship between both continuity notion can be found in [1, 3, 39].

The Moore-Yang integrals have two restrictions:

1. The interval function must be inclusion monotonic and

2. The limits of integration are real numbers.

Moore, Strother and Yang in  (pp. A-5) asseverate that:

“Theorem 4 (theorem  3.1 in this paper) suggests a more general definition for may be feasible – namely the right side of the equality in theorem 4. This conclusion could lead to a deletion of the condition .”

So, the first restriction, is not necessary and can be suppressed considering a more general definition of the integrals based on a characterization of the interval integral in term of an interval of real integrals. But, taken this definitions as primitive implies that the integral interval notion is not a generalization of a real integral approach, resulting in a notion with a poor mathematical foundation.

This paper define an integral for interval functions which extend the Moore-Yang approach eliminating both restrictions. We also give a characterization of this extension in terms of the extremes of the limits of integration which could simplify its computation.

Finally, we use this integral notion to determine probability density functions  in the context of interval probability [7, 30, 45].

## 2 Preliminary

Let be the set of real closed intervals, or simply intervals. We will use upper letters at start of alphabet to indicate an interval. The left and right extremes of an interval will be denoted by and respectively, thus . We define two projections for intervals: and .

The partial order that we will use for intervals will be the Kulisch-Miranker one , i.e.

 A≤B⇔a––≤b– and ¯¯¯a≤¯¯b.

Particularly,

 A≪B⇔a––

The partial order used in the Moore and Yang integral approach was the inclusion of sets, i.e.

 A⊆B⇔b–≤a––≤¯¯¯a≤¯¯b.

The arithmetical operations on intervals are defined as follow

 A⊕B={x⊕y∣x∈A and y∈B},

where is any one of the usual arithmetical operations. The unique restriction is that in the case of division, can not contain . Each operations can be characterized in terms of their extremes as follows:

 A⊕B=[min{a––⊕b–,a––⊕¯b,¯¯¯a⊕b–,¯¯¯a⊕¯¯b},max{a––⊕b–,a––⊕¯¯b,¯¯¯a⊕b–,¯¯¯a⊕¯¯b}].

For an abuse of language, given a real number , we will write and instead of and , respectively.

In the case of addition, this expression can be abbreviated as

 A+B=[a––+b–,¯¯¯a+¯¯b].

If either or then

 AB=[a––b–,¯¯¯a¯¯b].

Functions whose domain and co-domain are subsets of are called interval functions. Let be an interval function. Define the functions and by and . Trivially, . Sometimes will be denoted by . An interval function is said to be an inclusion monotonic function if it is monotonic w.r.to the inclusion, i.e. .

A distance between two intervals is defined by:

 dM(A,B)=max{∣b–−a––∣,∣¯¯b−¯a∣}.

In  was proved that is a metric. So, an interval function is continuous, if it is continuous w.r.to the metric .

## 3 Moore Approach

In this section we will overview the main definitions of the Moore and Yang interval integral approach.

###### Definition 3.1

Let be a real interval. A partition of is a sequence such that for each , . The set of all partition of will be denoted by .

###### Definition 3.2

Let be a real interval. A partition of is finer than the partition of , denoted by , if .

Clearly, is a partial order on .

###### Proposition 3.1

Let be a real interval. is a lattice with greatest element.

Proof: Let and be partitions of . Then trivially, and are the supremum and infimum, respectively, of and . The greatest element of is the partition .

###### Definition 3.3

Let be an interval and be an inclusion monotonic continuous interval function. The Riemann sum of w.r.to a partition of is defined by:

 ∑(F,T)=n−1∑k=0F([xk,xk+1])d(xk,xk+1),

where is the usual metric on the real numbers, i.e. .

The Moore and Yang integral of at the interval is defined by

 ∫AF(X)dX=⋂T∈T(A)∑(F,T).
###### Theorem 3.1

(Characterization theorem) Let be an interval and be an inclusion monotonic continuous interval function. Then

 ∫AF(X)dX=[∫¯¯¯aa––fl(x)dx,∫¯¯¯aa––fr(x)dx],

where and .

Proof: See .

## 4 Our Extension

###### Definition 4.1

Let and be two real intervals such that . Define,

 I[A,B]={X∈I(R)∣A≤X≤B}

and .

Since , then and . In what follows, without lost of generality, we will suppose that .

Given a metric space and a subset of we define the diameter of , denoted by , by

 diam(A)=sup{d(a,b)∣a,b∈A}.
###### Definition 4.2

A subset of a metric space is bounded if is finite

###### Definition 4.3

A function of a non-empty set into a metric space is called a bounded function if its image is a bounded set.

Clearly a subset of is bounded if, and only if, it is contained in for some such that . Thus, an interval function , where , is bounded if, and only if, its image is contained in for some . That is, if . Therefore, the topological and order based notions of boundedness for an interval function coincides.

Let be the function defined by, . Clearly the function is continuous and injective. Thus, we can identify with a subspace of , where is considered here with the metric,

 d((a––,¯¯¯a),(b–,¯¯b))=max{∣b–−a––∣,∣¯¯b−¯¯¯a∣}.

This is not restrictive since usual metrics in are equivalents, from the topological point-of-view.

Notice that is closed and bounded in , hence, by the Heine-Borel theorem  p. 119, it is compact.

###### Lemma 4.1

(Completeness Lemma) Let be a bounded subset of . Define and . Then, and where and .

Proof: If is a bounded subset of then for some . Clearly is a lower bound of and and is an upper bound of and . Since each upper bounded subset of the real numbers has a supremum and each lower bounded subset of the real numbers has an infimum, then and has supremum and infimum. It is easy to see that and .

###### Corollary 4.1

Let and be a bounded interval function. Then, the set has supremum and infimum. In fact, if then,

 ⨆F(I)=[⨆F––(I),⨆¯¯¯¯F(I)]

and

Proof: Since is a bounded interval function we have that for some . The statements follows from the Completeness lemma, by taking .

###### Lemma 4.2

Let and be bounded subsets of . Let

 I+J={X+Y∣X∈I,Y∈J}.

Then, and

Proof: The result follows by the proof of Lemma 4.1, by the characterization of the interval addition and by general properties of supremum and infimum.

###### Definition 4.4

A set is a partition of if there exists partitions and of and , respectively, such that .

###### Remark 4.1

Sometimes we say that the partition of cames from the partitions and of and , respectively, or simply, that of cames from .

###### Definition 4.5

We say that a partition of , is finer than the partition of , denoted by , if .

###### Lemma 4.3

Let is a partition of . Then is a lattice with greatest element.

Proof: Let and be partitions of coming from and , respectively. Then trivially, from definition of and proposition  3.1 we have that is the partition coming from . Analogously, we have that is the partition coming from . The greatest element of is the partition which came from .

###### Definition 4.6

Let be a bounded interval function. Given a partition of , we define the following Riemann sums of w.r.to namely:

• Lower Riemann sum –

• Upper Riemann sum –

 ∑(F,P)=n−1∑i=0m−1∑j=0⨆F(I[Pi,j,Pi+1,j+1])dM(Pi,j,Pi+1,j+1)

where and, as usual, .

###### Lemma 4.4

Let be a bounded function and let be a partition of . Then

 σ(F,P)≤∑(F,P).

Proof: Clearly, for each and , and . So,

 F(I[Pi,j,Pi+1,j+1])dM(Pi,j,Pi+1,j+1)≤⨆F(I[Pi,j,Pi+1,j+1])dM(Pi,j,Pi+1,j+1).

Therefore,

 σ(F,P)≤∑(F,P).

###### Proposition 4.1

Let be a bounded function and let and be partitions of . If then

 σ(F,P)≤σ(F,P′)≤∑(F,P′)≤∑(F,P).

Proof: Let be a partition cames from where . With no lost of generality we may assume that the partition cames from where .

Let then

 dM(Pi−1,j,Pi,j+1)≤dM(Pi−1,j,Qj)+dM(Qj+1,Pi,j+1).

On the other hand, we have that

Therefore, we have that

The second inequality was proved in Lemma  4.4 and the third inequality follows by the same token of the first.

###### Corollary 4.2

Let be a bounded function and let and be partitions of . Then,

 σ(F,P)≤Σ(F,Q).

Proof: Since the partition refines and we have that

 σ(F,P)≤σ(F,P∪Q)≤Σ(F,P∪Q)≤Σ(F,Q).

###### Definition 4.7

Let be a bounded function. We define the lower integral of w.r.to and , denoted by , by

 ∫BA––––F(X)dX=⨆P∈P[A,B]σ(F,P)

and the upper integral of w.r.to and , denoted by , by

###### Proposition 4.2

Let be a bounded function such that . Then, for any partition of we have that

 CdM(A,B)≤σ(F,P)≤∫BA––––F(X)dX≤¯¯¯¯¯¯¯¯¯∫BAF(X)dX≤∑(F,P)≤DdM(A,B).

Proof: Let be the trivial partition of . Then, by Lemma 4.1 we have that . But, by definition, we have that , which proves the first inequality. The last inequality follows by the same token.

The inequality follows by Corollary  4.2. The remaining inequalities follows by definition of the lower and upper integrals.

###### Proposition 4.3

Let and be subset of satisfying the following properties:

 (∙)∀P∈P[A,B]∃P′∈P′ and P′′∈P′′ such that σ(F,P)≤σ(F,P′) and ∑(F,P′′)≤∑(F,P).

Then,

 ∫BA––––F(X)dX=⨆P′∈P′σ(F,P′)

and

 ¯¯¯¯¯¯¯¯¯∫BAF(X)dX=P′′∈P′′∑(F,P′′).

Proof: This follows by general properties of supremum and infimum.

###### Corollary 4.3

Let , and let be the subset of consisting of partition containing . Then,

 ∫BA––––F(X)dX=⨆P∈P′[A,B]σ(F,P)

and

 ¯¯¯¯¯¯¯¯¯∫BAF(X)dX=P∈P′[A,B]∑(F,P).

Proof: From a partition build the partition , containing . Since is finer than we have that and . Thus, satisfies the condition of the above proposition.

###### Corollary 4.4

Let be the partition of coming from the partitions and of and , where and , respectively. Then,

 ∫BA––––F(X)dX=⨆n∈Nσ(F,Pn)

and

 ¯¯¯¯¯¯¯¯¯∫BAF(X)dX=n∈N∑(F,Pn).

Proof: Let be a partition of . Clearly, such that . Therefore, and . Thus, satisfies the condition

###### Definition 4.8

A bounded function is said to be an integrable function if

 ∫BA––––F(X)dX=¯¯¯¯¯¯¯¯¯∫BAF(X)dX.

This common value is called the interval integral of w.r.to and and it is denoted by .

###### Definition 4.9

Let be a bounded function. Define the left and right spectrum of , denoted by and respectively, by

 Fl(x)=π1F[x,¯¯¯a+¯¯b−¯¯¯ab–−a––(x−a––)]

and

 Fr(x)=π2F[x,¯¯¯a+¯¯b−¯¯¯ab–−a––(x−a––)],

where and are the left and right projections from to and .

###### Theorem 4.1

Let be a continuous function. Then,

 ∫BA––––F(X)dX=[∫b–a––––––Fl(x)dx,∫b–a––––––Fr(x)dx]dM(A,B)b–−a––

and

 ¯¯¯¯¯¯¯¯¯∫BAF(X)dX=⎡⎣¯¯¯¯¯¯¯¯¯¯∫–ba––Fl(x)dx,¯¯¯¯¯¯¯¯∫b–a––Fr(x)dx⎤⎦dM(A,B)b–−a––.

Proof: We will only prove the first equality since the second one follows analogously.

Let be the partition of coming from the partitions and of and , where and , respectively.

of , where Then,

 dM(Xk,Xk+1)=dM(A,B)n.

Therefore,

On the other side, we have that

 ∫b–a––––––Fl(x)dx=⨆n∈Nn−1∑k=0Fl(tk)d(tk,tk+1)=⨆n∈Nn−1∑k=0Fl(tk)(1n)(b–−a––)=(b–−a––)⨆n∈N(1n)n−1∑k=0F––(Xk).

Analogously we have that

 ∫b–a––––––Fr(x)dx=(b–−a––)⨆n∈N(1n)n−1∑k=0¯¯¯¯F(Xk).

Therefore, it is enough to prove that

 ⨆n∈N(1n)n−1∑k=0F––(I[Xk,Xk+1])=⨆n∈N(1n)n−1∑k=0F––(Xk)

and

 ⨆n∈N(1n)n−1∑k=0¯¯¯¯F(I[Xk,Xk+1])=⨆n∈N(1n)n−1∑k=0¯¯¯¯F(Xk).

By similarity, we will only prove the former.

Clearly we have that

 F––(I[Xk,Xk+1])≤F––(Xk).

Thus,

 ⨆n∈N(1n)n−1∑k=0F––(I[Xk,Xk+1])≤⨆n∈N(1n)n−1∑k=0F––(Xk).

Conversely, since is compact and is a continuous function, we have that such that .

Since is a continuous function and is compact we have that is uniformly continuous.

Let . Since is uniformly continuous, we have that such that if then .

Let be such that if then .

Since we have that

 dM(Xk,Yk)≤dM(Xk,Xk+1)=dM(A,B)n≤δ.

Thus, since is uniformly continuous, we have that . Therefore,

which proves that

 ⨆n∈N(1n)n−1∑k=0F––(I[Xk,Xk+1])=⨆n∈N(1n)n−1∑k=0F––(Xk).

###### Corollary 4.5

[characterization theorem] If is a continuous interval function then, is an integrable function and

 ∫BAF(X)dX=[∫b–a––Fl(x)dx,∫b–a––Fr(x)dx]d