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Moment Estimation Inequalities Based on Random Variable on Sugeno Measure Space
Journal of Inequalities and Applications volume 2010, Article number: 290124 (2010)
Abstract
The definitions and properties of moment of random variable are provided on Sugeno measure space. Then some important moment estimation inequalities based on random variable are presented and proven.
1. Introduction
In 1974, the Japanese scholar Sugeno [1] presented a kind of typical nonadditive measure, Sugeno measure, which is an important generalization of probability measure [2–6]. As we all know, the definitions and properties of moment of random variable play an important role in probability theory [7–9]. Likewise, they are also very important for Sugeno measure. In this paper we present the analogous definitions and properties based on random variable on Sugeno measure space. Then some important moment estimation inequalities based on random variable are presented and proven.
2. Preliminaries
Let us recall some definitions and facts from [5].
Definition 2.1.
Let be a nonempty set, let be a nonempty class of subsets of , and let be a nonnegative real valued set function defined on . Therefore satisfies the  rule (on ) if and only if there exists
such that
for any disjoint sequence of sets in whose union is also in .
Definition 2.2.
Let be a algebra of subsets of . And is called Sugeno measure on if and only if it satisfies the  rule and . Usually, Sugeno measure on is denoted by .
We call the triple a Sugeno measure space, denoted by space, where . In the following, our discussion will be restricted to this space.
Theorem 2.3.
For all imply that (monotonicity).
Theorem 2.4.
Let be a Sugeno measure on . Then, for any and ,
In order to present the analogous definitions and properties based on random variable on Sugeno measure space, we recall some definitions and facts from [10].
Definition 2.5.
Let be a function mapping from to real line . Then is called a random variable.
Definition 2.6.
Let be a random variable. Then the distribution function of is defined by
Definition 2.7.
Let be the distribution function of random variable . Then is called continuous random variable if there exists a nonnegative real valued function such that
is valid. The function is called a density function of .
In the following, our discussion will be restricted to the continuous random variable.
Definition 2.8.
Let be the distribution function of random variable . If , then we call an expected value of random variable , denoted by .
Theorem 2.9.
Let , be random variables; let C and D be constants. Then
Definition 2.10.
Let be a random variable. If exists, then is called the variance of , denoted by .
3. Moment Estimation Inequalities Based on Random Variable
We begin this section with a short lemma (see [11]), which will be useful in the sequel.
Lemma 3.1.
Let be a random variable whose Sugeno density function exists. If the Lebesgue integral
is finite, then
Theorem 3.2.
Let be a nonnegative random variable. When , the inequality
is valid; when , the inequality
holds true.
Proof.

(I)
When , since is a monotone decreasing function of we have
(3.5)

(II)
When , owing to the monotonicity of we also have
Definition 3.3.
Let be a random variable and a positive number. Then the expected value is called the th moment, the expected value is called the th absolute moment, the expected value is called the th central moment, and the expected value is called the th absolute central moment.
Theorem 3.4.
Let be a nonnegative random variable and a positive number. Then
Proof.
From Lemma 3.1, we infer
Similar to the case of credibility theory [12], we have the following: Theorems 3.5, 3.6, and 3.7.
Theorem 3.5.
Let be a random variable that takes values in and has expected value , and let be a convex function on . Then
Theorem 3.6.
Let be a random variable that takes values in and has expected value . Then
Theorem 3.7.
Let be a random variable that takes values in and has expected value . Then, for any positive integer ,
Theorem 3.8.
Let be a random variable and Then if and only if .
Proof.
From and Theorem 3.2, the conclusion is valid.
Theorem 3.9.
Let be a random variable and . If then Conversely, if there exists one positive number such that , then for any where
Proof.

(1)
When , we have
(3.12)
Since we obtain Consequently,
Since
we have

(2)
When , we have
Since
we obtain
Consequently,
Since
we have
Conversely, if then there exists one number such that , for all

(3)
When , for any where , we have
Since for any we have

(4)
When , for any where , we have
(3.24)
Since for any we have
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Acknowledgment
This work was supported by the NNSF of China (no. 60773062), the NSF of Hebei Province of China (no. 2008000633), the foundation of North China Electric Power University (no. 200911033), the KSRP of Department of Education of Hebei Province of China (no. 2005001D), and the KSTRP of Ministry of Education of China (no. 206012).
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Tian, J., Zhang, Z. & Tian, D. Moment Estimation Inequalities Based on Random Variable on Sugeno Measure Space. J Inequal Appl 2010, 290124 (2010). https://doi.org/10.1155/2010/290124
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DOI: https://doi.org/10.1155/2010/290124
Keywords
 Distribution Function
 Density Function
 Positive Integer
 Convex Function
 Measure Space