Some formulas of variance of uncertain random variable
Journal of Uncertainty Analysis and Applications
Some formulas of variance of uncertain random variable
Yuhong Sheng 1
Kai Yao 0
0 School of Management, University of Chinese Academy of Sciences , Beijing 100190 , China
1 Department of Mathematical Sciences, Tsinghua University , Beijing 100084 , China
Uncertainty and randomness are two basic types of indeterminacy. Chance theory was founded for modeling complex systems with not only uncertainty but also randomness. As a mixture of randomness and uncertainty, an uncertain random variable is a measurable function on the chance space. It is usually used to deal with measurable functions of uncertain variables and random variables. There are some important characteristics about uncertain random variables. The expected value is the average value of uncertain random variable in the sense of chance measure and represents the size of uncertain random variable. The variance of uncertain random variable provides a degree of the spread of the distribution around its expected value. In order to describe the variance of uncertain random variable, this paper provides some formulas to calculate the variance of uncertain random variables through chance distribution and inverse chance distribution. Several practical examples are also provided to calculate the variance for through chance distribution inverse chance distribution.
Chance theory; Uncertain random variable; Inverse chance distribution; Variance
Introduction
Probability theory is a branch of mathematics for studying the behavior of random
phenomena. In order to deal with randomness, Kolmogorov [
1
] defined a probability measure
as a set function satisfying the following three axioms: (i) normality, (ii) nonnegativity,
and (iii) additivity. Before applying it in practice, we should first obtain the probability
distribution via statistics or test the probability distribution to make sure it is close enough
to the real frequency, either of which is based on a lot of observed data. However, due to
the technological or economical difficulties, we sometimes have no observed data. In this
case, we have to rely on domain experts evaluating their belief degree about the chances
that the possible events happen. According to Kahneman and Tversky [
2
], human tends to
overweight unlikely events, so the belief degree generally has a much larger range than the
real frequency. As a result, the probability theory is not applicable in this case; otherwise,
it may lead to counterintuitive results [
3
]. Liu [
4
] gave some examples of it.
The uncertainty theory is a branch of mathematics for modeling belief degrees. In order
to deal with human uncertainty, Liu [
5
] first presented uncertain measure as a set
function satisfying four measure axioms: (i) normality, (ii) duality, (iii) subadditivity, and (iv)
product axiom [
6
]. As a fundamental concept in uncertainty theory, the uncertain variable
was presented by Liu [
5
] in 2007. In order to describe the uncertain variable, Liu [
5
]
introduced the concept of uncertainty distribution. Liu [
3
] proposed the concept of inverse
uncertainty distribution, and Liu [
4
] verified a sufficient and necessary condition for it.
After that, many researchers widely studied the uncertainty theory and made
significative progress. Gao [
7
] studied the properties of continuous uncertain measure. Peng and
Iwamura [
8
] proved a sufficient and necessary condition for uncertainty distribution.
Furthermore, a measure inversion theorem was given by Liu [
3
] that may yield uncertain
measures of some special events from the uncertainty distribution of the
corresponding uncertain variable. In addition, the concept of independence was proposed by Liu
[
6
]. After, Liu [
3
] presented the operational law of uncertain variables. In order to rank
uncertain variables, Liu [
5
] proposed the concept of expected value of uncertain variables.
The linearity of expected value operator was verified by Liu [
3
]. As an important
contribution, Liu and Ha [
9
] derived a useful formula for calculating the expected values of
strictly monotone functions of independent uncertain variables. Meanwhile, Liu [
5
]
presented the concept of variance of uncertain variables and also proposed some formulas
to calculate the variance through uncertainty distribution. Recently, Yao [
10
] proposed a
formula to calculate the variance using inverse uncertainty distribution. Sheng and Kar
[
11
] verified some results of moment of uncertain variable through inverse uncertainty
distribution.
However, in many cases, randomness and uncertainty exist simultaneously in a complex
system. In order to describe such a system, Liu [
12
] pioneered the concepts of uncertain
random variable and chance measure. Meanwhile, Liu [
12
] defined the concepts of chance
distribution, expected value and variance for uncertain random variable. Following that,
Liu [
13
] presented the operational l (...truncated)