The neutrosophic random variable can be expressed as

implying that for each fixed value of I N , X N is a deterministic linear function of the classical random variable X L . Hence, both share the same stochastic source and differ only by scale.

The key point is that the interval-valued quantity I N does not constitute a second stochastic mechanism in this formulation. Once I L and I U are fixed, every admissible value of X N is obtained from the same realization of X L by multiplication by a deterministic factor. Thus, the interval does not generate an additional random component; it specifies a range of deterministic transformations of the original random variable. An interval representation by itself, therefore, does not imply the presence of an additional probabilistic source of uncertainty.

Since I N varies deterministically within [ I L , I U ] , the neutrosophic random variable defines a bounded interval

This formulation ensures correct ordering regardless of the sign of X L . If X L ≥ 0 , the bounds appear as written; if X L < 0 , the order reverses automatically. In addition, the representation indicates that the entire neutrosophic interval is deterministically generated from the classical random variable X L through fixed proportional scaling, leaving no scope for independent randomness within the bounds.

Let X N L = ( 1 + I L ) X L and X N U = ( 1 + I U ) X L denote the lower and upper bounds of X N , respectively. These bounds are deterministic transforms of X L . When X L > 0 , the ordering is as written; when X L < 0 , the roles of the lower and upper bounds are reversed. These quantities are often treated as if they were independent or as if the interval [ X N L , X N U ] possessed its own probabilistic structure. However, both bounds are deterministic linear transformations of the same variable X L and therefore perfectly dependent.

The mathematical relationship between X N L and X N U is given by

implying Corr ( X N L , X N U ) = 1 . This perfect dependence is not merely a technical property of the two endpoints; it characterizes the entire interval construction. Once a realization of X L is specified, both endpoints are completely determined. Hence, observing one endpoint provides complete information about the other, apart from the fixed proportionality factor determined by I L and I U .

Consequently, the lower and upper bounds cannot be regarded as independent random quantities. The so-called neutrosophic interval merely represents a fixed proportional expansion of X L rather than a new layer of stochastic behavior. Treating these bounds as if they carried independent randomness or distinct inferential meaning is therefore mathematically inconsistent.

The width of the neutrosophic interval is defined as

For X L > 0 , this width is non-negative. If X L < 0 , the expression reverses sign, and the width should be taken as | ( I U − I L ) X L | to ensure non-negativity.

It follows immediately that W N is a deterministic multiple of the classical random variable X L . For fixed indeterminacy bounds I L and I U , the quantity W N inherits all its randomness from X L and does not constitute an additional stochastic component.

The ratio

is constant for all realizations of X L , indicating that the relative degree of “indeterminacy” is uniform across the entire support of the variable. Thus, the neutrosophic interval neither expands nor contracts in response to random variation; it merely scales linearly with X L .

Consequently, the width introduces no new uncertainty or informational content. It represents a fixed proportional transformation, reinforcing that the interval-valued construction remains entirely dependent on the classical random variable.

Let F X L ( x L ) and F X N ( x N ) denote the cumulative distribution functions (CDFs) of the classical random variable X L and the neutrosophic random variable X N , respectively.

For a neutrosophic random variable X N , we have

where I N ≥ 0 . Since I N ranges over [ I L , I U ] , the neutrosophic CDF of X N is interval-valued, given by

The interval representation of the CDF assumes X L > 0 . If X L < 0 , the ordering of the endpoints is reversed, and the CDF should be interpreted in terms of minimum and maximum values.

The representation of F X N ( x N ) in Equation (12) shows that the neutrosophic CDF is obtained directly from the classical CDF through deterministic scaling. Thus, the neutrosophic probability distribution is not distinct from the classical one; it is simply the classical CDF evaluated at scaled arguments. No new stochastic content is introduced by the interval-valued representation.

Let X L be a classical random variable with a location-scale probability density function (PDF): f X L ( x ; θ L ) , where θ L may denote a vector of underlying location and scale parameters. Recall that the X N can be expressed as X N = ( 1 + I N ) X L for I N ∈ [ I L , I U ] , where I N is a fixed but interval-valued indeterminacy.

Under this linear transformation, the neutrosophic parameters θ N are simple scaled versions of the classical parameters θ L , with the scaling factor determined by I N . For example, if X L follows a normal distribution with mean μ L and variance σ L 2 ; then, for any fixed value

of I N , the neutrosophic normal distribution has the mean μ N = ( 1 + I N ) μ L and the variance σ N 2 = ( 1 + I N ) 2 σ L 2 . Analogous scaling occurs for other location-scale families, such as exponential, gamma, and uniform distributions.

For a fixed I N ∈ [ I L , I U ] , the PDF of X N can be written as

where θ L = θ N / ( 1 + I N ) is the corresponding classical parameter vector.

For a random sample x N = ( x N , 1 , ⋯ , x N , n ) , the neutrosophic likelihood function is given by

or

where x L = x N / ( 1 + I N ) is the corresponding classical sample.

Since this holds for every I N ∈ [ I L , I U ] , the neutrosophic likelihood for the interval-valued random variable is proportional to the classical likelihood across the entire indeterminacy range:

The factor ( 1 + I N ) − n is a Jacobian term arising solely from the deterministic change of scale. It does not introduce a new component into the likelihood. Once the observations and parameters are transformed back to the classical scale, the remaining likelihood is exactly the classical likelihood. Thus, the apparent difference between the two likelihoods is attributable to parameterization and scale, rather than to a different probabilistic mechanism.

Therefore, likelihood-based analyses—including likelihood ratios, score functions, and maximum likelihood estimation—are fully determined by the classical model once the scaling is accounted for. The interval-valued indeterminacy does not introduce any new inferential content or independent stochastic variation, confirming that the neutrosophic likelihood is merely a scaled version of the classical likelihood.

Let X L denote a classical random variable with characteristic function (CF) defined as

Let X N = ( 1 + I N ) X L denote the corresponding neutrosophic random variable, where I N ∈ [ I L , I U ] is interval-valued indeterminacy.

The CF of X N is given by

Since I N ranges over [ I L , I U ] , the neutrosophic CF is a set given by:

Thus the neutrosophic CF is simply the classical CF evaluated at scaled arguments. Because the CF uniquely determines the distribution, X N and X L are distributionally equivalent up to deterministic scaling. The indeterminacy I N does not introduce new randomness or informational content; it merely rescales the argument of the classical CF.

Let X L be a classical random variable with mean μ L and variance σ L 2 .

The skewness and kurtosis of X L , denoted by γ 1 , L and γ 2 , L , are given by

respectively. For the neutrosophic random variable X N = ( 1 + I N ) X L with I N ∈ [ I L , I U ] , the mean and variance are μ N = ( 1 + I N ) μ L and σ N 2 = ( 1 + I N ) 2 σ L 2 , respectively.

Then the skewness and kurtosis of X N are

respectively. Since these equalities hold for all I N ∈ [ I L , I U ] , we conclude

Therefore, the skewness and kurtosis of the neutrosophic random variable are identical to those of the classical random variable across the entire indeterminacy interval. This confirms that neutrosophic scaling does not modify distributional shape, further reinforcing the complete dependence of X N on X L .

3.8.1. Discrete Random Variable

Let X L be a discrete (classical) random variable with the PMF P L ( X L = x L ) . The Shannon entropy of X L is given by

For the neutrosophic random variable X N = ( 1 + I N ) X L with I N ∈ [ I L , I U ] , the PMF satisfies

Since each x N corresponds uniquely to x L = x N / ( 1 + I N ) , the set of probabilities { P L ( x L ) } is preserved. Therefore,

for all I N ∈ [ I L , I U ] . Thus, the Shannon entropy of the neutrosophic random variable is identical to that of the classical random variable, confirming that interval-valued indeterminacy introduces no additional uncertainty in the discrete case.

3.8.2. Continuous Random Variable

Let X L be a continuous (classical) random variable with the PDF f X L ( x L ) .

Its differential Shannon entropy is

For the neutrosophic random variable X N = ( 1 + I N ) X L with I N ∈ [ I L , I U ] , the PDF is given by

The entropy of X N becomes

Since I N ∈ [ I L , I U ] , the neutrosophic entropy is interval-valued:

The additive constant log ( 1 + I N ) arises solely from deterministic scaling and does not represent new uncertainty. Hence, neutrosophic entropy is merely a shifted version of the classical entropy, confirming redundancy in the continuous case as well.

To illustrate the redundancy of interval-valued indeterminacy, consider

Mean: E [ X N ] ∈ [ 11 , 13 ] , obtained by scaling E [ X L ] = 10 .Variance: V ( X N ) ∈ [ 4.84 , 6.76 ] , obtained by scaling V ( X L ) = 4 .CDF: F X N ( 15 ) ∈ [ F X L ( 11.54 ) , F X L ( 13.64 ) ] , illustrating that the neutrosophic CDF is obtained from the classical CDF through deterministic scaling.Entropy: H ( X N ) ∈ [ H ( X L ) + log ( 1.1 ) ,   H ( X L ) + log ( 1.3 ) ] , a deterministic shift of the classical entropy.Skewness/Kurtosis: identical to classical values (0 and 3).

These calculations confirm that neutrosophic measures are merely interval-scaled versions of classical measures, introducing no new stochastic content.

To illustrate the interval-valued nature of the neutrosophic random variable, consider a classical sample drawn from X L ~ N ( μ L = 10 ,   σ L 2 = 1 ) :

Let the indeterminacy vary over I N ∈ [ 0.1 , 0.3 ] . Then the parameters of the neutrosophic normal distribution are μ N ∈ [ 11 , 13 ] and σ N 2 ∈ [ 1.21 , 1.69 ] .

For each observation x L , i , for i = 1 , 2 , ⋯ , 5 , the neutrosophic random variable is

Explicitly, the neutrosophic sample is given by

The classical MLEs are

The neutrosophic MLEs are interval-scaled versions:

These MLEs can be verified as

This example confirms that each neutrosophic observation is simply a deterministic interval scaling of the classical sample, and that neutrosophic estimators are scaled versions of their classical counterparts. Moreover, the transformation is reversible: since the indeterminacy bounds are fixed and known, each original classical observation can be recovered exactly from either endpoint as X i = X L , i / ( 1 + I L ) = X U , i / ( 1 + I U ) . Thus, the neutrosophic sample contains the same underlying information as the classical sample, expressed in interval form. No new randomness or inferential content is introduced by the interval-valued indeterminacy. Similar numerical examples can be found in Aslam [11] and Aslam [12].

More importantly, the example illustrates a general relationship rather than merely a numerical coincidence. Once the classical observations are given, every endpoint of every neutrosophic observation is determined. Conversely, when the fixed indeterminacy interval is known, the original classical observations can be recovered exactly from either endpoint. Thus, the mapping between the classical and neutrosophic observations is one-to-one and involves only deterministic scaling. Consequently, the neutrosophic representation does not provide a different statistical analysis; it merely expresses the same classical analysis in interval form.