International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 7888
Published online March 25, 2022
https://doi.org/10.5391/IJFIS.2022.22.1.78
© The Korean Institute of Intelligent Systems
Ahmad M. H. Alkhazaleh^{1} and Shawkat Alkhazaleh^{2 }
^{1}Department of Mathematics, Faculty of Science, AlAlbayt University, AlMafraq, Jordan
^{2}Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan
Correspondence to :
Shawkat Alkhazaleh (shmk79@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution NonCommercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Data could be uncertain, and the levels of precision of data are intuitively different. Neutrosophic set expressions are considered an alternative to represent imprecise data in such cases. In this paper, a general definition of neutrosophic conditional probability is introduced as a generalization of the classical conditional probability. Additionally, the properties of this neutrosophic conditional probability are presented. The concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function in the classical type are generalized to a neutrosophic type with two discrete and continuous neutrosophic random variables. Various properties and examples are presented to demonstrate the significance of this study.
Keywords: Conditional probability, Neutrosophic conditional probability, Neutrosophic distribution function, Marginal neutrosophic density function, Neutrosophic expected value, Joint neutrosophic density function
Crisp is the most important requirement in classical mathematics, whereas real problems involve uncertain data. Thus, the solution to these problems involves the use of mathematical principles based on uncertainty (not crisp). Therefore, many scientists and engineers have been interested in uncertainty modeling to describe and debrief useful information hidden in uncertain data. To help them deal with this uncertainty, numerous theories such as the fuzzy set theory [1], intuitionistic fuzzy set theory [2], rough set theory [3], and neutrosophic set theory [4,5] have been proposed in recent years.
Smarandache proposed the theory of neutrosophic logic as a general framework for the unification of many existing logics, such as the intuitionistic fuzzy logic. This theory aims to be a new mathematical tool for handling problems involving imprecise, indeterminant, and inconsistent data. The main idea of neutrosophic logic is to distinguish each logical statement in a threedimensional neutrosophic space, wherein each dimension of the space represents the truth
The neutrosophic probability given by Smarandache [6], which is a generalization of the classical and imprecise probabilities in which the chance that an event
Although the neutrosophic probability theory is one of the most important tools and has applications in real life, it has not received significant attention. However, it has been the focus of some studies. For more information about neutrosophic probability, see [6–8].
In 2003, Smarandache [9], for the first time, introduced the notions of neutrosophic measure and neutrosophic integral. Neutrosophic measure is a generalization of the classical measure when the space contains some indeterminacy, and the neutrosophic integral is defined on the neutrosophic measure. Hanafy et al. [10–13] studied the correlation coefficient under uncertainty. Thereafter, Salama et al. [14] in 2014, introduced and studied the concepts of correlation and correlation coefficient of neutrosophic data in probability spaces and some of their properties. In addition, they introduced and studied the neutrosophic simple linear regression model and provided a possibility of its application to data processing. By applying the neutrosophic probability in physics, Yuhua [15] in 2015, determined the neutrosophic probability of accelerating the expansion of the partial universe. Some problems and solutions related to the neutrosophic statistical distribution, given by Patro and Smarandache [16] in 2016 and Smarandache et al. [17] in 2017, used proportional conflict redistribution rule number 5 (PCR5) to combine the information of two sources providing subjective probabilities of an event A occurring with a chance that A occurs, an indeterminate chance that A occurs, and a chance that A does not occur. Likewise, in 2017, Guo et al. [18] proposed an evidence fusion method based on neutrosophic probability analysis in the DSmT framework. They also introduced some basic theories, including DST, DSmT, and the dissimilarity measure of evidence. Consequently, in 2017, Gafar and ElHenawy [19] presented a framework of ant colony optimization and entropy theory and used it to define a neutrosophic variable from concrete data. In their paper, they exhibited the incorporation of a hybrid search model amongst ant colony optimization and information theory measures to demonstrate a neutrosophic variable. Taking a new step towards the study of neutrosophic probabilities in 2018, Alhabib et al. [20] introduced and studied some neutrosophic probability distributions by generalizing some classical probability distributions such as the Poisson distribution, exponential distribution, and uniform distribution to the neutrosophic type. Subsequently, in 2019, Alhasan and Smarandache studied the neutrosophic Weibull distribution and the Weibull family along with the relationship of the functions with the neutrosophic Weibull—such as the inverse Weibull, Rayleigh distribution, threeparameter Weibull, beta Weibull, five Weibull, and six Weibull distributions under the neutrosophic case. A general definition of neutrosophic random variables was introduced by Zeina and Hatip [21] in 2021. They studied the properties of this concept and generalized the probability distribution function, cumulative distribution function, expected value, variance, standard deviation, mean deviation, rth quartiles, moment generating function, and characteristic function from crisp logic to neutrosophic logic. In this paper, as a generalization of the classical conditional probability, we introduce a general definition of neutrosophic conditional probability and its properties. In addition, we will generalize, from the classical type to the neutrosophic type, the concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function. We do this using two neutrosophic random variables, discrete and continuous. The significance of this study is demonstrated by providing numerous properties and examples.
In this section, we recall the definitions that are related to this work. The neutrosophic set, neutrosophic probability, and neutrosophic random variables are defined.
Let
where
Classical neutrosophic number has the form
The neutrosophic probability of an event
where
A neutrosophic random (stochastic) variable is subject to change due to both randomness and indeterminacy, while the classical random (stochastic) variable is subject to change only due to randomness. The values of this variable represent the possible outcomes and possible indeterminacies. Randomness and indeterminacy can be either objective or subjective.
A neutrosophic random variable is a variable that may have an indeterminate outcome.
A neutrosophic random (stochastic) process represents the evolution of some neutrosophic random values over time. This is a collection of random neutrosophic variables.
Consider the crisp random variable
Consider the neutrosophic random variable
Consider the neutrosophic random variable
Consider the neutrosophic random variable
Properties of the expected value of a neutrosophic random variable.
If

Consider the neutrosophic random variable
Smarandache [9] discussed neutrosophic conditional probability by comparing it with classical probability. In classical probability, if
where the neutrosophic Bayesian rule is
In this section, we introduce the concepts of the neutrosophic distribution function, neutrosophic regular conditional probabilities, and neutrosophic marginal density function. The properties of these concepts were proved, and some examples were obtained.
Let (Ω,ℑ,
Let
the neutrosophic regular conditional probabilities are defined as
Let
where
For the neutrosophic marginal density function of
where
Now,
Letting Δ(
This relation shows that the function
is the neutrosophic conditional density of
Following are some theorems related to expected value of a neutrosophic random variable:
(Linearity expected of two neutrosophic random variables).
Continuous
Discrete
(Multiplication expected of two neutrosophic random variables).
Continuous
Discrete
In classical probability,
Continuous
Discrete
For any two neutrosophic random variables
Continuous
Discrete
In classical probability, two discrete random variables
Two discrete neutrosophic random variables
Equivalently,
If
Continuous
Discrete
If
If
If
If
If
Let
We can then write a joint neutrosophic density function as follows:
For 0
where
To prove that the density function
We can find the marginal
To prove its PDF, let
We can also find the marginal
To prove its PDF, let
Let
Then we can write a joint neutrosophic density function as follows:
We can find the marginal
To prove its PDF, let
We can also find the marginal
To prove its PDF, let
Conditional probabilities are an important topic that contribute to solving numerous everyday problems. Many researchers have studied these probabilities under uncertain conditions. In this paper, conditional probabilities are studied, for the first time, under the neutrosophic theory. We introduced a neutrosophic conditional probability as a generalization of the classical conditional probability, and presented its properties. The concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function in the classical type are generalized with the neutrosophic type using discrete and continuous cases. Numerous properties and examples are provided to demonstrate the significance of this study. We suggest that researchers use these results and apply them to bivariate distribution probabilities. We also suggest expanding these results to the concept of nvalued neutrosophic logic.
No potential conflict of interest relevant to this article was reported.
Email: ahmed 2005kh@yahoo.com
Email: shmk79@gmail.com
International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 7888
Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.78
Copyright © The Korean Institute of Intelligent Systems.
Ahmad M. H. Alkhazaleh^{1} and Shawkat Alkhazaleh^{2 }
^{1}Department of Mathematics, Faculty of Science, AlAlbayt University, AlMafraq, Jordan
^{2}Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan
Correspondence to:Shawkat Alkhazaleh (shmk79@gmail.com)
This is an Open Access article distributed under the terms of the Creative Commons Attribution NonCommercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Data could be uncertain, and the levels of precision of data are intuitively different. Neutrosophic set expressions are considered an alternative to represent imprecise data in such cases. In this paper, a general definition of neutrosophic conditional probability is introduced as a generalization of the classical conditional probability. Additionally, the properties of this neutrosophic conditional probability are presented. The concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function in the classical type are generalized to a neutrosophic type with two discrete and continuous neutrosophic random variables. Various properties and examples are presented to demonstrate the significance of this study.
Keywords: Conditional probability, Neutrosophic conditional probability, Neutrosophic distribution function, Marginal neutrosophic density function, Neutrosophic expected value, Joint neutrosophic density function
Crisp is the most important requirement in classical mathematics, whereas real problems involve uncertain data. Thus, the solution to these problems involves the use of mathematical principles based on uncertainty (not crisp). Therefore, many scientists and engineers have been interested in uncertainty modeling to describe and debrief useful information hidden in uncertain data. To help them deal with this uncertainty, numerous theories such as the fuzzy set theory [1], intuitionistic fuzzy set theory [2], rough set theory [3], and neutrosophic set theory [4,5] have been proposed in recent years.
Smarandache proposed the theory of neutrosophic logic as a general framework for the unification of many existing logics, such as the intuitionistic fuzzy logic. This theory aims to be a new mathematical tool for handling problems involving imprecise, indeterminant, and inconsistent data. The main idea of neutrosophic logic is to distinguish each logical statement in a threedimensional neutrosophic space, wherein each dimension of the space represents the truth
The neutrosophic probability given by Smarandache [6], which is a generalization of the classical and imprecise probabilities in which the chance that an event
Although the neutrosophic probability theory is one of the most important tools and has applications in real life, it has not received significant attention. However, it has been the focus of some studies. For more information about neutrosophic probability, see [6–8].
In 2003, Smarandache [9], for the first time, introduced the notions of neutrosophic measure and neutrosophic integral. Neutrosophic measure is a generalization of the classical measure when the space contains some indeterminacy, and the neutrosophic integral is defined on the neutrosophic measure. Hanafy et al. [10–13] studied the correlation coefficient under uncertainty. Thereafter, Salama et al. [14] in 2014, introduced and studied the concepts of correlation and correlation coefficient of neutrosophic data in probability spaces and some of their properties. In addition, they introduced and studied the neutrosophic simple linear regression model and provided a possibility of its application to data processing. By applying the neutrosophic probability in physics, Yuhua [15] in 2015, determined the neutrosophic probability of accelerating the expansion of the partial universe. Some problems and solutions related to the neutrosophic statistical distribution, given by Patro and Smarandache [16] in 2016 and Smarandache et al. [17] in 2017, used proportional conflict redistribution rule number 5 (PCR5) to combine the information of two sources providing subjective probabilities of an event A occurring with a chance that A occurs, an indeterminate chance that A occurs, and a chance that A does not occur. Likewise, in 2017, Guo et al. [18] proposed an evidence fusion method based on neutrosophic probability analysis in the DSmT framework. They also introduced some basic theories, including DST, DSmT, and the dissimilarity measure of evidence. Consequently, in 2017, Gafar and ElHenawy [19] presented a framework of ant colony optimization and entropy theory and used it to define a neutrosophic variable from concrete data. In their paper, they exhibited the incorporation of a hybrid search model amongst ant colony optimization and information theory measures to demonstrate a neutrosophic variable. Taking a new step towards the study of neutrosophic probabilities in 2018, Alhabib et al. [20] introduced and studied some neutrosophic probability distributions by generalizing some classical probability distributions such as the Poisson distribution, exponential distribution, and uniform distribution to the neutrosophic type. Subsequently, in 2019, Alhasan and Smarandache studied the neutrosophic Weibull distribution and the Weibull family along with the relationship of the functions with the neutrosophic Weibull—such as the inverse Weibull, Rayleigh distribution, threeparameter Weibull, beta Weibull, five Weibull, and six Weibull distributions under the neutrosophic case. A general definition of neutrosophic random variables was introduced by Zeina and Hatip [21] in 2021. They studied the properties of this concept and generalized the probability distribution function, cumulative distribution function, expected value, variance, standard deviation, mean deviation, rth quartiles, moment generating function, and characteristic function from crisp logic to neutrosophic logic. In this paper, as a generalization of the classical conditional probability, we introduce a general definition of neutrosophic conditional probability and its properties. In addition, we will generalize, from the classical type to the neutrosophic type, the concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function. We do this using two neutrosophic random variables, discrete and continuous. The significance of this study is demonstrated by providing numerous properties and examples.
In this section, we recall the definitions that are related to this work. The neutrosophic set, neutrosophic probability, and neutrosophic random variables are defined.
Let
where
Classical neutrosophic number has the form
The neutrosophic probability of an event
where
A neutrosophic random (stochastic) variable is subject to change due to both randomness and indeterminacy, while the classical random (stochastic) variable is subject to change only due to randomness. The values of this variable represent the possible outcomes and possible indeterminacies. Randomness and indeterminacy can be either objective or subjective.
A neutrosophic random variable is a variable that may have an indeterminate outcome.
A neutrosophic random (stochastic) process represents the evolution of some neutrosophic random values over time. This is a collection of random neutrosophic variables.
Consider the crisp random variable
Consider the neutrosophic random variable
Consider the neutrosophic random variable
Consider the neutrosophic random variable
Properties of the expected value of a neutrosophic random variable.
If

Consider the neutrosophic random variable
Smarandache [9] discussed neutrosophic conditional probability by comparing it with classical probability. In classical probability, if
where the neutrosophic Bayesian rule is
In this section, we introduce the concepts of the neutrosophic distribution function, neutrosophic regular conditional probabilities, and neutrosophic marginal density function. The properties of these concepts were proved, and some examples were obtained.
Let (Ω,ℑ,
Let
the neutrosophic regular conditional probabilities are defined as
Let
where
For the neutrosophic marginal density function of
where
Now,
Letting Δ(
This relation shows that the function
is the neutrosophic conditional density of
Following are some theorems related to expected value of a neutrosophic random variable:
(Linearity expected of two neutrosophic random variables).
Continuous
Discrete
(Multiplication expected of two neutrosophic random variables).
Continuous
Discrete
In classical probability,
Continuous
Discrete
For any two neutrosophic random variables
Continuous
Discrete
In classical probability, two discrete random variables
Two discrete neutrosophic random variables
Equivalently,
If
Continuous
Discrete
If
If
If
If
If
Let
We can then write a joint neutrosophic density function as follows:
For 0
where
To prove that the density function
We can find the marginal
To prove its PDF, let
We can also find the marginal
To prove its PDF, let
Let
Then we can write a joint neutrosophic density function as follows:
We can find the marginal
To prove its PDF, let
We can also find the marginal
To prove its PDF, let
Conditional probabilities are an important topic that contribute to solving numerous everyday problems. Many researchers have studied these probabilities under uncertain conditions. In this paper, conditional probabilities are studied, for the first time, under the neutrosophic theory. We introduced a neutrosophic conditional probability as a generalization of the classical conditional probability, and presented its properties. The concepts of joint distribution function, regular conditional probabilities, marginal density function, expected value, and joint density function in the classical type are generalized with the neutrosophic type using discrete and continuous cases. Numerous properties and examples are provided to demonstrate the significance of this study. We suggest that researchers use these results and apply them to bivariate distribution probabilities. We also suggest expanding these results to the concept of nvalued neutrosophic logic.
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