Abstract
Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.
Highlights
Many problems in real life exist that are full of confusion, vagueness, and ambiguity. e quantification of such issues in a scientific way is the need of time. e negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. e distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. e literature lacks in dealing with the situations for interval-valued data under negative binomial distribution
Order statistics for the proposed distribution, including wth, joint, median, minimum, and maximum order statistics are part of the paper. e proposed distribution is discussed from the real data applications perspective by considering the different case studies. is research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously
Introduction e term neutrosophy was introduced by Smarandache [1], a modern philosophical branch inspired by famous fuzzy logic
Summary
Rehan Ahmad Khan Sherwani, Sadia Iqbal, Shumaila Abbas, Muhammad Aslam ,2 and Ali Hussein AL-Marshadi. Interval-valued fuzzy sets were introduced by [12] Smarandache claimed both neutrosophic sets and neutrosophic statistics to generalize fuzzy logic [13, 14]. E classical negative binomial distribution is generalized neutrosophically, which ensures some indeterminacy related to the probabilistic experiment. E neutrosophic negative binomial random variable is defined as a variable number of trials to obtain the fixed number of successes. It is known as a neutrosophic negative binomial distribution. . .}, Npr (occurrences of a fixed number of successes for a variable number of trials) (Tx, Ux, Ix), probability mass function and cumulative distribution function of neutrosophic negative binomial distribution are, respectively, given as. Physical Conditions (i) Each trial results in three mutually exclusive and exhaustive outcomes such as success, failure, and indeterminacy (ii) All the trials must be independent (iii) e probability of success remains fixed or constant for each trial (iv) An experiment is repeated a variable number of times to produce a fixed number of successes
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