probability density function

We apply the Riemann-Liouville fractional integral to generalize a companion of Ostrowski's type integral inequality. The present article recaptures all the results of M. W. Alomari's article and also for one more article of different authors. Applications are also deduced for numerical integration, probability theory and special means.

The concept of Z-number was introduced by Zadeh in order to deal with partial reliability of information. This concept describes a fusion of fuzzy and probabilistic types of uncertainty. In turn, one of the main fields of dealing with imperfect information is approximate reasoning. For the case of pure fuzzy information this field is well-developed. In contrast, existing studies on reasoning with Z-valued “if-then” rules are scarce. One of the main reasons is high analytical and computational complexity. In this work, we develop an approach to reasoning with such kind of rules. The original approach proposed here allows to deal with sparse rule base and is characterized by relatively low computational complexity. The new concept of similarity of Z-numbers based on Jaccard similarity index and measure of divergence of probability distributions is introduced. Based on similarity degrees of current input Z-numbers and Z-numbers located in rule antecedents, weights of linear combination of Z-numbers in rule consequents are determined. The linear combination is based on operations with Z-numbers proposed by authors. Applications of the proposed approach to evaluation of economic development level for a country and control problem are considered. 

This study introduces the Choquet integral of fuzzifying functions with respect to a fuzzy measure. To express various phenomena or ambiguous values in many applications may not be enough to show them as a function, in which case a fuzzifying function can be applied to achieve better expressions or flexibility for given function values. To apply Choquet integrals of fuzzifying functions, we consider Choquet integrals of interval-valued functions as an operator which are α-level functions of fuzzifying functions. In this study, we investigate some properties of Choquet integral of fuzzifying functions and present their applications. As part of this, a series of relevant examples and their subsequent applications are provided, along with the fuzzification of two integrands: a probability density function (PDF) and a utility function.

The purpose of the present study is to consider the estimation of the PDF and CDF of the three-parameter inverse Weibull (IWD) distribution. To do so, we propose the following well-known methods moment (MM) estimation, maximum likelihood (ML) estimation, and a developed method entitled the location and scale parameters free maximum likelihood (LSPF) derived from Nagatsuka et al. (2013). Having estimated the parameters, we would consider estimating the PDF and the CDF of the IWD distribution with these three methods. Then, analytical expressions are derived for the mean integrated squared error (MISE) to compare the estimators. According to the results of simulation and two real data for estimation of the PDF and CDF, when the shape parameter is greater than 1, the LSPF method performs better than the others, and when the shape parameter is equal to or smaller than 1, the ML method is better than the others.

The purpose of this article is to generalize Cebysev type inequalities for double integrals involving a weight function.By using an integral transform that is a weighted Montgomery identity, we obtained a generalized form of weighted Cebysev type inequalities in $L_m,, mgeq 1$ norm of differentiable functions. Also, we give some applications of the probability density function.