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The Z-number deserves further exploration in uncertain environments to effectively address fuzziness and reliability in practical issues simultaneously. Based on the credibility distribution and conversion rules, we define its expected value, variance, and semi-variance, and demonstrate the feasibility of these calculations by deriving formulas. Moreover, we explore three characteristics inherent in symmetrical Z-numbers. The link between the variance and semi-variance of Z-numbers is discovered and proved. Furthermore, we apply the formulas for expected value and variance to Z-numbers in examples, the results of which validate our proposed formulas. The findings underscore the significance of our study in applying the expected value and variance of fuzzy sets across diverse fields.

In a chess match, one player plays with white pieces while the other plays with black. It is generally known that players playing with the white pieces have a slight advantage [1]. To keep it fair, a tournament tries to have each player play an equal number of games with the white pieces as with the black pieces. However, in a single round-robin, this is not achievable. Each player would have some rounds where they play with white and some rounds with black. Further, they might have consecutive rounds where they play with the same colour. This creates a psychological challenge, particularly if a player has to play with black pieces in consecutive rounds late in a round-robin tournament. This article introduces a new metric that counts the total pairs of rounds when any player plays with the same color. Then, a construction of how each player should play with another player in every round of a single or double round-robin tournament that minimizes this sum is provided. This will help to ensure maximum fairness to the players in round-robin chess tournaments. [ABSTRACT FROM AUTHOR]

Abstract We give new inequalities involving the expectation and variance of a random variable defined on a finite interval and having an essentially bounded probability density function. Our inequalities sharpen previous inequalities by N.S. Barnett and S.S. Dragomir. We prove our bounds are optimal, and we explicitly give the cases in which equality occurs.

This paper explains the dilemma of artificial intelligence in relation to the development of teacher education based on the functional structure of artificial intelligence and the activity characteristics of teacher education. Then, after designing a survey questionnaire on the factors affecting the development of teacher education empowered by artificial intelligence and completing the reliability test, the paper collects initial data in the form of distributing questionnaires and analyzes in detail the least squares estimation of mean, variance, standard deviation, correlation coefficient, and regression coefficient needed in the process of analyzing the data to carry out the analysis of instances. The correlation coefficients of teacher training, professional development, policy support, resource allocation, teacher literacy, educational information technology behaviors, and AI-enabled teacher education development are 0.674 (0.003), 0.496 (0.001), 0.259 (0.009), 0.371 (0.008), 0.639 (0.004), and 0.325 (0.007). Their corresponding regression coefficients were 0.616 (t=59.852, P=0.003), 0.021 (t=0.018, P=0.007), 0.078 (t=5.668, P=0.005), 0.032 (t=3.282, P=0.009), 0.239 (t=29.734, P=0.008), 0.137 (t=5.406, P=0.001), indicating that these factors have a significant impact relationship on AI-enabled teacher education.

An iterative algorithm for identifying unknown parameters of a mathematical model based on the Bayesian approach is proposed, which makes it possible to determine the most probable maximum informative estimates of these parameters. The example of the mathematical model of mass transfer dynamics shows the algorithm for finding the most probable and most informative estimate of the vector of unknown parameters, and also an analysis of the sequence of the corresponding steps is given. The results of computational experiments showed a significant dependence of the results of the calculations on the choice of the initial approximation point and slowing down the rate of convergence of the iterative process (and even its divergence) with an unsuccessful choice of the initial approximation. The validity of the obtained results is provided by analytical conclusions, the results of computational experiments, and statistical modeling. The results of computational experiments make it possible to assert that the proposed algorithm has a sufficiently high convergence for a given degree of accuracy and makes it possible to derive not only estimates of point values of mathematical model parameters based on a posteriori analysis, but also confidence intervals of these estimates. At the same time, it should be noted that the results of calculations depend significantly on the choice of the initial approximation point and the slowing of the convergence rate of the iterative process with an unsuccessful choice of the initial approximation. Analytical studies and results of calculations confirm the effectiveness of the proposed identification algorithm, which makes it possible, with the help of active, purposeful experiments, to build more accurate mathematical models. In accordance with the algorithm, a program was developed in the MatLab mathematics package and computational experiments were performed.

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial.

The aim of this paper is to develop new nonparametric estimators of entropy based on the kth nearest neighbor distances that are considered between n sample points, k ≤ (n − 1) being a positive integer, fixed. The Method consists in using the new estimators which were useful in order to evaluate the entropies for random vectors. As results, using the Kaniadakis entropy measure, the asymptotic unbiasedness and consistency of the estimators are proven.

The trustworthiness of mobile nodes is considered as the principal parameter for ensuring significant data dissemination in a Mobile Ad hoc NETwork (MANET). However, the selfish behaviour of nodes minimizes the trust by dropping a considerable amount of data packets in the network. The significant dropping of data packets by the selfish node introduces huge data overhead with increased latency and energy consumption, thus increasing the number of retransmissions. In this paper, Selfish Node Detection Scheme based on Bates Distribution Inspired Trust Factor (SNDS-BDITF) is propounded for predominant detection of selfish behaviour by investigating multiple levels of factors that contribute towards effective selfishness detection. The proposed SNDS-BDITF approach is also potent in enhancing the detection rate of selfishness by multi-perspective analysis of each monitored node’s forwarding characteristics considering the benefits of other cooperating mobile nodes. The simulation results of the propounded SNDS-BDITF method are enhanced on an average by 16% and 14% when compared to the existing selfish node segregation mechanisms prevalent in the literature.

The article presents new tools for investigating the statistical properties of the harmonic signal autocorrelation function (ACF). These tools enable identification of the ACF estimator errors in measurements in which the triggering of the measurements is non-synchronized. This is important because in many measurement situations the initial phase of the measured signal is random. The developed tools enable testing the ACF estimator of a harmonic signal in the presence of Gaussian noise. These are the formulas on the basis of which the statistical properties of the estimator can be determined, including the bias, the variance and the mean squared error (MSE). For comparison, the article also presents the ACF statistical analysis tools used in the conditions of synchronized measurement triggering, known from the literature. Operation of the new tools is verified by simulation and experimental studies. The conducted research shows that differences between the MSE results obtained with the use of the developed formulas and those attained from simulations and experimental tests are not greater than 1 dB.

In this paper, we detail closed form estimators for beta distribution that are simpler than those proposed by Tamae, Irie and Kubokawa. The proposed estimators are shown to have smaller asymptotic variances and smaller asymptotic covariances compared to Tamae estimators and maximum likelihood estimators. The proposed estimators are also shown to perform better in real data applications.

In this paper, general definition of neutrosophic random variables is introduced and its properties are presented. Concepts of probability distribution function, cumulative distribution function, expected value, variance, standard deviation, mean deviation, r th quartiles, moments generating function and characteristic function in crisp logic are generalized to neutrosophic logic. Many solved problems and applications are presented which show the power of the study and show the ability of applying the results in various domains including quality control, stochastic modeling, reliability theory, queueing theory, decision making, electrical engineering, … etc.

Some extensions of the results related to bounds for the eigenvalues and spreads of matrices using traces are discussed here. We also obtain bounds for the eigenvalues of a positive definite matrix in terms of its trace and trace of its inverse. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce one discrete random variable, namely the negative λ-binomial random variable. We deduce the expectation of the negative λ-binomial random variable. We also get the variance and explicit expression for the moments of the negative λ-binomial random variable.

Abstract In this study, random linear difference equations obtained by transforming the components of deterministic difference equations to random variables are investigated. Uniform, Bernoulli, binomial, negative binomial (or Pascal), geometric, hypergeometric and Poisson distributions have been used for the random effects for obtaining the random behavior of linear difference equations. The random version of the Z-transform, the RZ-transform, has been used to obtain an approximation for the random linear difference equation. Approximate expected values and variances are calculated by using the RZ-transform. The results have been obtained with Maple and are shown in graphs. It is shown that the random Z-transform is an effective tool for the investigation of random linear difference equations.

In statistical analysis, oftentimes a probability density function is used to describe the relationship between certain unknown parameters and measurements taken to learn about them. As soon as there is more than enough data collected to determine a unique solution for the parameters, an estimation technique needs to be applied such as “fractional calculus”, for instance, which turns out to be optimal under a wide range of criteria. In this context, we aim to present some novel estimates based on the expectation and variance of a continuous random variable by employing generalized Riemann-Liouville fractional integral operators. Besides, we obtain a two-parameter extension of generalized Riemann-Liouville fractional integral inequalities, and provide several modifications in the Riemann-Liouville and classical sense. Our ideas and obtained results my stimulate further research in statistical analysis.

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research.

Interval estimation is an important topic, especially in drawing conclusions on an event. Mathematics education students must possess the skill to formulate and use interval estimation. The errors of mathematics education students in formulating wrong interval estimates indicate a low understanding of interval estimation. This study explores the errors of mathematics education students in interpreting the variance in the questions regarding selecting the proper test statistic to formulate the interval estimation of mean accurately. Respondents in this study involved 36 students of mathematics education (N = 9 males, N = 27 females). This research is qualitative research with a qualitative descriptive approach. Data collection was carried out using the respondents’ ability test and interviews. The respondents’ ability test instrument was tested on 36 students and declared valid where r-count > r-table with r-table of 0.3291, and declared reliable with a Cronbach Alpha value of 0.876 > 0.6. Through an exploratory approach, data were analyzed by categorizing, reducing, and interpreting to conclude students' abilities and thinking methods in formulating interval estimation of the mean based on the variance in questions. The results showed that mathematics education students neglected the variance, so they could not determine the test statistics correctly, resulting in error interval estimates. This study provides insight into the thinking methods of mathematics education students on variance in interval estimation problems in the hope of anticipating errors in formulating interval estimation problems.

The integral of a smooth function with bounded support over a set with finite perimeter in Euclidean space ℝd is estimated using a periodic grid in an isotropic uniform random position. Extension term in the estimator variance is proportional to the integral of the squared modulus of the function over the object boundary and to the grid scaling factor raised to the power of d+1. Our result generalizes the Kendall-Hlawka-Matheron formula for the variance of the isotropic uniform systematic estimator of volume.

The kurtosis of the distribution of financial returns characterized by high volatility persistence and thick tails is notoriously difficult to estimate precisely. We propose a simple but effective procedure of estimating the kurtosis coefficient (and variance) based on volatility filtering that uses a simple GARCH model. In addition to an estimate, the proposed algorithm issues a signal of whether the kurtosis (or variance) is finite or infinite. We also show how to construct confidence intervals around the proposed estimates. Simulations indicate that the proposed estimates are much less median biased than the usual method-of-moments estimates, their confidence intervals having much more precise coverage probabilities. The procedure alsoworks well when the underlying volatility process is not the one the filtering technique is based on. We illustrate how the algorithm works using several actual series of returns.

This article studies the role of model uncertainties in sensitivity and probability analysis of reliability. The measure of reliability is failure probability. The failure probability is analysed using the Bernoulli distribution with binary outcomes of success (0) and failure (1). Deeper connections between Shannon entropy and variance are explored. Model uncertainties increase the heterogeneity in the data 0 and 1. The article proposes a new methodology for quantifying model uncertainties based on the equality of variance and entropy. This methodology is briefly called “variance = entropy”. It is useful for stochastic computational models without additional information. The “variance = entropy” rule estimates the “safe” failure probability with the added effect of model uncertainties without adding random variables to the computational model. Case studies are presented with seven variants of model uncertainties that can increase the variance to the entropy value. Although model uncertainties are justified in the assessment of reliability, they can distort the results of the global sensitivity analysis of the basic input variables. The solution to this problem is a global sensitivity analysis of failure probability without added model uncertainties. This paper shows that Shannon entropy is a good sensitivity measure that is useful for quantifying model uncertainties.

Fatigue in a material occurs when it is subjected to fluctuating stress and strain, which usually results in failure due to the accumulated damage. In statistics, asymmetric distribution, which is commonly used for describing the fatigue life of materials, is the Birnbaum–Saunders (BS) distribution. This distribution can be transform to the normal distribution, which is symmetrical. Furthermore, variance is used to examine the dispersion of the fatigue life data. However, comparing the variances of two independent samples that follow BS distributions has not previously been reported. To accomplish this, we propose methods for providing the confidence interval for the ratio of variances of two independent BS distributions based on the generalized fiducial confidence interval (GFCI), a Bayesian credible interval (BCI), and the highest posterior density (HPD) intervals based on a prior distribution with partial information (HPD-PI) and a proper prior with known hyperparameters (HPD-KH). A Monte Carlo simulation study was carried out to examine the efficacies of the methods in terms of their coverage probabilities and average lengths. The simulation results indicate that the HPD-PI performed satisfactorily for all sample sizes investigated. To illustrate the efficacies of the proposed methods with real data, they were also applied to study the confidence interval for the ratio of the variances of two 6061-T6 aluminum coupon fatigue-life datasets.

The objective of this article is to develop the link between probability distribution theory and subdivision schemes. The distribution theory is a field of statistics and subdivision is a field of Computer Aided Geometric Design, both are independent fields. The analysis of subdivision schemes using probability distribution theory is presented. In our analysis, we explore the important properties such as mean, variance, moments about the origin, moments about mean, measures of skewness and measures of kurtosis of subdivision schemes presented in Chinese Annals of Mathematics, Series B (8(5): 1077-1092, 2017). [ABSTRACT FROM AUTHOR]

In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields.

Motivated by applications in electron microscopy, we study the situation where a stationary and isotropic random field is observed on two parallel planes with unknown distance. We propose an estimator for this distance. Under the tractable, yet flexible class of Lévy-based random field models, we derive an approximate variance of the estimator. The estimator and the approximate variance perform well in two simulation studies.

The currently performed mathematical and computer modeling of thermal processes in technical systems is based on an assumption that all the parameters determining thermal processes are fully and unambiguously known and identified (i.e., determined). Meanwhile, experience has shown that parameters determining the thermal processes are of undefined interval-stochastic character, which in turn is responsible for the intervalstochastic nature of thermal processes in the electronic system. This means that the actual temperature values of each element in an technical system will be randomly distributed within their variation intervals. Therefore, the determinative approach to modeling of thermal processes that yields specific values of element temperatures does not allow one to adequately calculate temperature distribution in electronic systems. The interval-stochastic nature of the parameters determining the thermal processes depends on three groups of factors: (a) statistical technological variation of parameters of the elements when manufacturing and assembling the system; (b) the random nature of the factors caused by functioning of an technical system (fluctuations in current and voltage; power, temperatures, and flow rates of the cooling fluid and the medium inside the system); and (c) the randomness of ambient parameters (temperature, pressure, and flow rate). The interval-stochastic indeterminacy of the determinative factors in technical systems is irremediable; neglecting it causes errors when designing electronic systems. A method that allows modeling of unsteady interval-stochastic thermal processes in technical systems (including those upon interval indeterminacy of the determinative parameters) is developed in this paper. The method is based on obtaining and further solving equations for the unsteady statistical measures (mathematical expectations, variances and covariances) of the temperature distribution in an technical system at given variation intervals and the statistical measures of the determinative parameters. Application of the elaborated method to modeling of the interval-stochastic thermal process in a particular electronic system is considered.

Abstract Variance is of great significance in measuring the degree of deviation, which has gained extensive usage in many fields in practical scenarios. The definition of the variance on the basis of the credibility measure was first put forward in 2002. Following this idea, the calculation of the accurate value of the variance for some special fuzzy variables, like the symmetric and asymmetric triangular fuzzy numbers and the Gaussian fuzzy numbers, is presented in this paper, which turns out to be far more complicated. Thus, in order to better implement variance in real-life projects like risk control and quality management, we suggest a rational upper bound of the variance based on an inequality, together with its calculation formula, which can largely simplify the calculation process within a reasonable range. Meanwhile, some discussions between the variance and its rational upper bound are presented to show the rationality of the latter. Furthermore, two inequalities regarding the rational upper bound of variance and standard deviation of the sum of two fuzzy variables and their individual variances and standard deviations are proved. Subsequently, some numerical examples are illustrated to show the effectiveness and the feasibility of the proposed inequalities.

A shift-dependent information measure is favorable to handle in some specific applied contexts such as mathematical neurobiology and survival analysis. For this reason, the weighted differential entropy has been introduced in the literature. In accordance with this measure, we propose the weighted generalized cumulative residual entropy as well. Despite existing apparent similarities between these measures, however, there are quite substantial and subtle differences between them because of their different metrics. In this paper, particularly, we show that the proposed measure is equivalent to the generalized cumulative residual entropy of the cumulative weighted random variable. Thus, we first provide expressions for the variance and the new measure in terms of the weighted mean residual life function and then elaborate on some characteristics of such measures, including equivalent expressions, stochastic comparisons, bounds, and connection with the excess wealth transform. Finally, we also illustrate some applications of interest in system reliability with reference to shock models and random minima.

In this paper, an attempt has been made to develop a periodic review inventory model by considering lead-time and the backorder rate as control variables in fuzzy stochastic environment. Without loss of generality, we have assumed that all the observed values of the fuzzy random variable, representing the demand as triangular fuzzy numbers. The variance of fuzzy random demand is taken into consideration to give due attention to every fuzzy observations. The protection interval demand has also been assumed to be fuzzy stochastic. The expected shortages are calculated by using credibility criterion. For the proposed model, we provide a solution procedure incorporating numerical technique viz. Scan and zoom method to determine an optimal policy. A numerical example is taken up to illustrate the solution procedure and sensitivity analysis of the optimal solution with respect to the key parameters of the system is carried out.

A novel estimator for estimating the mean length of fibres is proposed for censored data observed in square shaped windows. Instead of observing the fibre lengths, we observe the ratio between the intensity estimates of minus-sampling and plus-sampling. It is well-known that both intensity estimators are biased. In the current work, we derive the ratio of these biases as a function of the mean length assuming a Boolean line segment model with exponentially distributed lengths and uniformly distributed directions. Having the observed ratio of the intensity estimators, the inverse of the derived function is suggested as a new estimator for the mean length. For this estimator, an approximation of its variance is derived. The accuracies of the approximations are evaluated by means of simulation experiments. The novel method is compared to other methods and applied to real-world industrial data from nanocellulose crystalline.

Control charts are considered as powerful tools in detecting any shift in a process. Usually, the Shewhart control chart is used when data follows the symmetrical property of a normal distribution. In practice, the data from the industry may follow a non-symmetrical distribution or an unknown distribution. The average run length (ARL) is a significant measure to assess the performance of the control chart. The ARL may mislead when the statistic is computed from an asymmetric distribution. To handle this issue, in this paper, an ARL-unbiased hybrid exponentially weighted moving average proportion (HEWMA-p) chart is proposed for monitoring the process variance for a non-normal distribution or an unknown distribution. The efficiency of the proposed chart is compared with the existing chart in terms of ARLs. The proposed chart is more efficient than the existing chart in terms of ARLs. A real example is given for the illustration of the proposed chart in the industry.

In the present paper, the variance of local stereological volume estimators is studied. For isotropic designs, the variance depends on the shape of the body under study and the choice of reference point. It can be expressed in terms of an equivalent star body. For a collection of triaxial ellipsoids the variance is determined by simulation. The problem of estimating particle size distributions from central sections through the particles is also discussed.

We obtained the gonad volume and oocyte number in a single Venerupis pullastra (Montagu, 1803) using unbiased stereological estimators. Since the gonad is not a fixed anatomical structure and is merged with other organs in the clam, we used new variance expressions to study the efficiency of these estimators, where the isotropy condition is not required for the covariogram.

Periodic spatial grids can be used for unbiased estimation of length and surface area of objects by counting or measuring intersections of the objects with the grids. The estimators are theoretically based on discrete approximation of well established integral geometric formulas. The variance of the estimates depends on properties of both the grid and the measured objects. Main results of the theory of variance of the isotropic uniform random (IUR) volume estimation by spatial grids, especially a formula relating the variance of the volume estimator with the object surface area and the grid constant, are recapitulated. To identify main features of length and surface area IUR estimates the variance due to rotation and simple asymptotic formulas for the residual variance of estimates of selected model objects is calculated. Surface area estimates by multiple grids of parallel lines in 3D and of the variance of length estimates by periodic grids of planes or spheres in ddimensional space are studied.

In the present paper the statistical properties of local stereological estimators of particle volume are studied. It is shown that the variance of the estimators can be decomposed into the variance due to the local stereological estimation procedure and the variance due to the variability in the particle population. It turns out that these two variance components can be estimated separately, from sectional data. We present further results on the variances that can be used to determine the variance by numerical integration for particular choices of particle shapes.

While in multi-user environment, CPU has to manage lot of requests generated over the same time. Waiting queue of processes generates a problem of scheduling for processors. Designers and hardware architects have suggested system of multiprocessors to overcome the queue length. Lottery scheduling is one such method where processes in waiting queue are selected through a chance manner. This opens a way to use probability models to get estimates of system parameters. This paper is an application where the processing time of jobs in ready queue is predicted using the sampling method under the k-processors environment (k>1).The random selection of one process by each of k processors through without replacement method is a sample data set which helps in the prediction of possible ready queue processing time. Some theorems are established and proved to get desired results in terms of confidence intervals.

A latent variable modeling procedure for examining specificity in any indicator of a common factor for a set of repeatedly administered measures is outlined in a longitudinal design setting with measurement invariance and specificity stability over time. The method permits one to test whether there is specificity in a given indicator and in the affirmative case allows one to point and interval estimate the specificity variance in that measure. The discussed procedure is readily applicable with popular software, is based on empirically testable conditions, and is illustrated with an example. [ABSTRACT FROM PUBLISHER]

In this paper we estimate that the equation of smoothing to the first Smoothing is the same equation of double exponentioal smoothing, and finding the equation of prediction of two state first: the variance of the original time cerise xt ,( σ2xt )is the same as the variance of the Ft, , (σ2Ft), ( σ2xt=, σ2Ft), and the second state the variance of xt , , (σ2xt)is not the same as the variance of Ft , σ2ft , ( σ2xt=, σ2Ft),. Ft This results will be applied to the real data which represent the mean of relative wet of Mosul city, and also to the data obtained form simulation, with use the (χ2 , MSD) to select the best of smoothing.