Narrow Results

Fuzzy variables are functions from credibility spaces to the set of real numbers. The set of fuzzy variables is a linear space with the classic operations of addition and multiplication by numbers. Its subspace formed by fuzzy variables with finite p^th absolute moments is showed to be a complete para-normed space. The concept of para-normed space is novel, and is an extension of normed space. It is seen that most properties of normed spaces hold in para-normed spaces. Also some useful inequalities in para-normed spaces are obtained.

Generalizing a first approach by L. A. ZADEH (J. Math. Anal. Appl. 23, 1968), expected values of fuzzy events are studied which are (up to standard boundary conditions) only required to be monotone. They can be seen as an extension of capacities, i.e., monotone set functions satisfying standard boundary conditions. Some of these expected values can be characterized axiomatically, others are based on some distinguished integrals (Choquet, Sugeno, Shilkret, universal, and decomposition integral).

Interval multi-objective linear programming (IMOLP) ímodels are one of the methods to tackle uncertainties. In this paper, we propose two methods to determine the efficient solutions in the IMOLP models through the expected value, variance and entropy operators which have good properties. One of the most important properties of these methods is to obtain different efficient solutions set according to decision makers’ preferences as available information. We first develop the concept of the expected value, variance and entropy operators on the set of intervals and study some properties of the expected value, variance and entropy operators. Then, we present an IMOLP model with uncertain parameters in the objective functions. In the first method, we use the expected value and variance operators in the IMOLP models and then we apply the weighted sum method to convert an IMOLP model into a multi-objective non-linear programming (MONLP) model. In the second method, the IMOLP model using the expected value, variance and entropy operators can be converted into a multi-objective linear programming (MOLP) model. The proposed methods are applicable for large scale models. Finally, to illustrate the efficiency of the proposed methods, numerical examples and two real-world models are solved.

Possibility, necessity and credibility measures are used in the literature in order to deal with imprecision. Recently, Yang and Iwamura [L. Yang and K. Iwamura, Applied Mathematical Science2(46) (2008) 2271–2288] introduced a new measure as convex linear combination of possibility and necessity measures and they determined some of its axioms. In this paper, we introduce characteristics (parameters) of a fuzzy variable based on that measure, namely, expected value, variance, semi-variance, skewness, kurtosis and semi-kurtosis. We determine some properties of these characteristics and we compute them for trapezoidal and triangular fuzzy variables. We display their application for the determination of optimal portfolios when assets returns are described by triangular or trapezoidal fuzzy variables.

The following sections are included:

• INTRODUCTION

• DISCRETE AND CONTINUOUS RANDOM VARIABLES

• PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES
  • Probability Distribution
  • Probability Function and Cumulative Distribution Function

• EXPECTED VALUE AND VARIANCE FOR DISCRETE RANDOM VARIABLES
  • Expected Value for Earnings per Share
  • Expected Value of Ages of Students
  • Expected Value and Variance: Defective Tires
  • Expected Value and Variance: Commercial Lending Rate

• THE BERNOULLI PROCESS AND THE BINOMIAL PROBABILITY DISTRIBUTION
  • The Bernoulli Process
  • Binomial Distribution
  • Probability Function
  • Mean and Variance

• THE HYPERGEOMETRIC DISTRIBUTION
  • The Hypergeometric Formula
  • Mean and Variance

• THE POISSON DISTRIBUTION AND ITS APPROXIMATION TO THE BINOMIAL DISTRIBUTION
  • The Poisson Distribution
  • The Poisson Approximation to the Binomial Distribution

• JOINTLY DISTRIBUTED DISCRETE RANDOM VARIABLES
  • Joint Probability Function
  • Marginal Probability Function
  • Conditional Probability Function
  • Independence

• EXPECTED VALUE AND VARIANCE OF THE SUM OF RANDOM VARIABLES
  • Covariance and Coefficient of Correlation Between Two Random Variables
  • Expected Value and Variance of the Summation of Random Variables X and Y
  • Expected Value and Variance of Sums of Random Variables

• Summary

• Appendix 6A The Mean and Variance of the Binomial Distribution

• Appendix 6B Applications of the Binomial Distribution to Evaluate Call Options
  • What Is an Option?
  • The Simple Binomial Option Pricing Model
  • The Generalized Binomial Option Pricing Model¹⁰

• Questions and Problems

The following sections are included:

• PROBABILITY
• EXPECTATIONS
• DISTRIBUTIONS
• STATISTICAL ESTIMATION
• STATISTICAL TESTING
• DATA TYPES
• PROBLEM SET
• FURTHER RECOMMENDED READINGS