Statistical Modeling and Analysis using Doob’s Theorem in Continuous Distributions

Prepared by the researche  : Maria Mohammed Baher ¹, Ruqaya Shaker Mahmood ², Mohammed RASHEED ^{2, 3, *}

• ¹ Department of Business Administration, Ibn Khaldun private university college, Baghdad, Iraq
• ² Applied Sciences Department, University of Technology- Iraq, Baghdad, Iraq
• ³ Laboratoire Moltech Anjou Universite d’Angers/UMR CNRS 6200, 2, Bd Lavoisier, 49045 Angers, France

DAC Democratic Arabic Center GmbH

Journal of Afro-Asian Studies : Twenty-sixth Issue – August 2025

A Periodical International Journal published by the “Democratic Arab Center” Germany – Berlin

:To download the pdf version of the research papers, please visit the following link

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Abstract

Doob’s theorem, a cornerstone of martingale theory, provides critical insights into understanding stochastic processes and continuous distributions. This research highlights both its theoretical foundation and practical relevance through three interconnected examples. The first explores its application in Gaussian signal processing and machine learning, where the theorem’s principles of stationarity and independence enable accurate modeling and prediction within these systems.

In the second example, financial modeling leverages Geometric Brownian Motion (GBM) to simulate stock price movements. By applying martingale theory, the theorem ensures fair valuation of options in uncertain market conditions. This approach is instrumental in managing financial risk and informing decision-making processes.

The third case applies stochastic processes grounded in Doob’s theorem to environmental science, specifically analyzing temperature and rainfall interactions. This method uncovers significant relationships among climatic variables, offering valuable insights for ecological studies and climate forecasting.

By connecting theoretical probability with real-world applications, this research demonstrates how Doob’s theorem supports statistical modeling, prediction, and the understanding of complex stochastic systems. Its utility in fields like finance and environmental science underscores its importance for addressing practical challenges. Through a combination of theoretical concepts and computational simulations, this study showcases the wide-ranging relevance of Doob’s theorem in contemporary research.

1. Introduction

Doob’s theorem serves as a cornerstone of martingale theory, offering a robust framework for understanding stochastic processes and continuous distributions [1-3]. It plays a critical role in fields such as finance, environmental science, and operations research, where complex probabilistic systems must be analyzed [4-6]. By establishing the stationarity and independence of martingales through appropriate filtrations, Doob’s theorem provides analysts and researchers with the tools needed to develop effective statistical models for continuous distributions [7-10].

This paper explores both the theoretical foundations and practical applications of Doob’s theorem, using numerical examples and computer simulations to demonstrate its relevance. Gaussian processes, vital in many statistical applications, are examined first [11-13]. Doob’s theorem ensures their stationarity and independence, enabling accurate modeling [14, 15]. In finance, Geometric Brownian Motion is applied to simulate stock price dynamics, highlighting martingale properties critical for pricing financial derivatives and understanding market behavior [16].

In environmental science, stochastic processes based on Doob’s theorem are used to study the relationships between temperature and rainfall, providing insights into ecological interdependencies [17-19]. Queueing theory, a core aspect of operations research, benefits from the theorem’s ability to model waiting times and assess system performance [20, 21]. Similarly, reliability engineering uses Doob’s theorem to estimate system lifespans and failure probabilities, essential for minimizing downtime and ensuring operational continuity [22-24].

This study demonstrates how Doob’s theorem bridges theoretical probability and practical application, enhancing statistical modeling across diverse fields. By offering solutions to challenges in finance, environmental science, and operations research, Doob’s theorem underscores its versatility in addressing real-world problems through advanced stochastic modeling

2. Doob’s Theorem (DT)

DT affects stochastic processes and continuous distributions in martingale theory [25]. Certain circumstances may make a stochastic process martingale, allowing investigation of its behavior and features [26]. Consider a stochastic process  defined on filtered probability space , where  represents information up to time . Doob’s theorem asserts that  is a martingale if and only if it meets three requirements [27-30].

Doob’s theorem plays a pivotal role in martingale theory, significantly influencing stochastic processes and continuous distributions [31]. Under specific conditions, a stochastic process can qualify as a martingale, enabling a deeper exploration of its properties and behavior [32]. To illustrate, consider a stochastic process ​ defined on a filtered probability space , where ​ encapsulates all the information available up to time t [33, 34]. According to Doob’s theorem, the process  is a martingale if and only if it satisfies three essential criteria [35]. These conditions serve as the foundation for determining whether a stochastic process exhibits the martingale property, providing a powerful tool for analyzing dynamic systems [36-40].

1. Adaptedness: _​ is -measurable for all .
2. Integrability: for all .
3. Martingale Property: For all _s​

Mathematically, the martingale property can be expressed as:

(1)

The optional stopping and decomposition theorems improve Doob’s usefulness. The optional stopping theorem states that if  fits certain criteria, then . Fintech uses Doob’s theorem for option pricing, while environmental modeling uses it to explain stochastic variable dependencies [41, 42].

This study utilized Doob’s theorem to analyze and characterize stochastic processes across three key areas: Gaussian processes, financial modeling through Geometric Brownian Motion (GBM), and the environmental modeling of temperature and rainfall relationships.

1. Gaussian Processes (Example 1)

Doob’s theorem proved Gaussian process stationarity and independence [43-45]. The martingale attribute was tested to ensure the process’s estimated future value matched its present value based on earlier observations [46-48]. Verifying the simulated Gaussian process’s theoretical behavior required this [50, 51]. The theory mathematically supported stationarity tests like the Augmented Dickey-Fuller (ADF) test and autocorrelation analysis [52, 53].

Step 1: Define the Gaussian Process

We consider a Gaussian process () defined by , where  are i.i.d. random variables. The goal is to test stationarity and independence using Doob’s theorem [54].

Step 2: Verify the Martingale Property

To confirm that () satisfies the martingale property:

(2)

Since ϵt​ are i.i.d. with , we compute:

 (3)

Thus, () is a martingale.

Step 3: Stationarity Test

We test stationarity using the Augmented Dickey-Fuller (ADF) and KPSS tests:

• ADF test checks the presence of a unit root, indicating non-stationarity.
• KPSS test examines whether the series is stationary around a deterministic trend.

The calculated p-values showed stationarity (p<0.05 for ADF) and confirmed that (Y_t) meets the stationary condition [55-58].

Step 4: Independence Test

The autocorrelation function (ACF) was computed for lags 1,2,…,51, 2, …, 51,2,…,5. The ACF values were close to zero and not significant, confirming the independence of ().

Gaussian Processes: Demonstrated stationarity and independence, confirming the process as a martingale

Gaussian Process Data: This dataset will include 1,000 observations of a Gaussian process () simulated as ​, where .

2. Financial Modeling with GBM (Example 2)

In financial modeling, Doob’s theorem played a pivotal role in simulating stock price dynamics using GBM [21]. The martingale property of GBM ensured that the expected discounted future stock price equaled the current price, a fundamental assumption in option pricing. This allowed for accurate and fair valuation of financial derivatives under the risk-neutral measure. Doob’s theorem provided the theoretical guarantee that GBM maintained the necessary martingale structure, enabling effective financial decision-making and risk management [22].

Step 1: Define GBM for Stock Prices

Stock prices (S_t) are modeled as [56]:

(4)

where: S₀​ is the initial stock price, μ is the drift rate, σ is the volatility and W_t is a standard Brownian motion.

Step 2: Validate the Martingale Property

Under the risk-neutral measure, the drift rate becomes μ=r (risk-free rate), ensuring that:

(5)

To convert to a martingale, we use the discounted stock price:

(6)

Now:

(7)

This confirms  is a martingale.

Step 3: Simulate Stock Prices

Using simulated parameters (=100, =0.05, =0.2, =1, =252), stock prices were generated over 252 trading days. The final stock price, mean, standard deviation, and minimum/maximum values were computed.

Step 4: Analyze Log Returns

Log returns  were calculated, and their ACF values were analyzed to confirm that returns exhibit no significant autocorrelation, aligning with the martingale property.

Financial Modeling: Validated the martingale property in GBM and analyzed simulated stock prices and log returns.

Geometric Brownian Motion (GBM) Data: This dataset will include simulated stock prices over 252 trading days based on the GBM model:

  with parameters: =100, =0.05, =0.2, and =1 year.

3. Environmental Modeling (Example 3)

For environmental modeling, Doob’s theorem helped establish and analyze the stochastic relationship between temperature and rainfall [57]. The theorem guided the construction of a martingale-based model, ensuring that the stochastic process capturing these dependencies adhered to essential probabilistic properties [25]. This improved ecology and climatic research by revealing how temperature affects rainfall patterns [26]. These examples showed Doob’s theorem’s adaptability in verifying and modeling real-world stochastic processes, connecting abstract theory to practical applications [27].

Step 1: Define the Stochastic Process

​ and ​ indicate temperature and rainfall at t-time, respectively. A linear stochastic relationship is modeled as [58]:

(8)

where , , , and

Step 2: Verify the Martingale Property

Using Doob’s theorem, we ensure  is a martingale:

(9)

this condition applies because  is independent noise.

Step 3: Estimate Parameters

Based on regression analysis,  and  were calculated. The residuals  were verified for independence and stationarity using ACF and stationarity tests.

Step 4: Simulate Temperature and Rainfall Dependencies

Estimated parameters simulated temperature and rainfall. The stochastic model was supported by the high correlation coefficient (=0.76).

Environmental Modeling: Given a stochastic link between temperature and rainfall using martingale-based residual analysis. These calculations demonstrate Doob’s theorem’s practical use in verifying and modeling stochastic processes in several domains.

Environmental Modeling Data: The link between temperature () and rainfall  will be simulated in this dataset.

3. Experimental Methods

Numerical examples using simulation and statistical modeling demonstrate Doob’s theorem. Each example illustrates Doob’s theorem-related characteristics of continuous distributions. The datasets and statistical tests have been simulated to prove the stationarity and independence of Gaussian processes derived from Doob’s theorem [59, 60].

Furthermore these qualities using actual data. A geometric Brownian motion has been used and validated to generate sample routes for stock prices and calculate martingale measures to improve option pricing models and analyze market volatility in financial modeling.

Environmental Stochastic Modeling: We use regression approaches based on Doob’s theorem to investigate temperature and rainfall connections over time.

4. Results and Discussion
   • The Gaussian Process Example

Analyses showed that simulated Gaussian processes are stationary and independent. Doob’s theorem predictions were confirmed by statistical tests with p-values over the significance level. In this example, the properties of Gaussian processes using Doob’s theorem to establish stationarity and independence has been investigated. We simulate a Gaussian process and apply statistical tests to verify these properties.

Simulation Setup: We simulate a univariate Gaussian process with mean zero and variance one over a discrete time interval. The process is generated using the following equation [32]:

(10)

where () are independent normally distributed random variables .

Table 1 presents the summary statistics of the simulated Gaussian process, demonstrating its central tendency and dispersion

Table 1: Summary Statistics of the Gaussian Process

From Table 1 the summary statistics of the simulated Gaussian process, as presented in Table 1, reveal key insights into its behavior. The mean of 0.002 is very close to zero, aligning with the theoretical expectation for a process generated with a mean of zero. The standard deviation of 0.95 is close to one, supporting the normal distribution variance assumption. The lowest and highest values, -3.19 and 3.42, demonstrate large oscillations, characteristic of Gaussian random variations. Finally, the dataset has 1,000 observations, ensuring statistical dependability. These results corroborate the simulation’s theoretical features, making it suitable for stochastic modeling and statistical analysis. Stationarity tests for the simulated Gaussian process are shown in Table 2.

Table 2: Results of Stationarity Tests

The stationarity tests in Table 2 strongly support the simulated Gaussian process’s stationarity. The Augmented Dickey-Fuller (ADF) test statistic was -12.56 with a p-value below 0.05. Rejecting the null hypothesis of non-stationarity proves the process is stationary. Similar to the KPSS test, the p-value was 0.23, over the significance level. This indicates we cannot reject the stationarity null hypothesis. These two complimentary tests indicate that the Gaussian process is stationary, which is necessary for stochastic modeling and statistical analysis. The simulated Gaussian process time series plot is shown in Fig. 1. The variations around the zero mean show the process’s unpredictability and volatility over time.

Fig. 1. Simulated Gaussian process time series plot

Fig. 1 shows the simulated Gaussian process’s time series with random variations around zero. Its stationarity is confirmed by its lack of trend. The process follows a Gaussian distribution since the values fluctuate within the predicted standard deviation. Lack of systematic patterns emphasizes process unpredictability, making it suited for stochastic phenomena modeling. The simulated Gaussian process autocorrelation function (ACF) is shown in Fig. 2. The quick decrease approaches 0 suggests process values at various time delays are independent

Fig. 2: Autocorrelation Function (ACF)

For different time delays, Fig. 2 shows the Gaussian process autocorrelation function (ACF). The quick decline of ACF values to near zero and their lack of statistical significance imply observation independence. Uncorrelated process values at distinct time points are essential to Gaussian processes. The ACF plot validates the theoretical expectation of no persistent correlations, confirming the approach for stochastic and statistical models with independent increments.

ACF findings for the first five lags of the Gaussian process are shown in Table 3, showing no substantial autocorrelation.

Table 3. Gaussian process ACF values for the first five lags

The simulated Gaussian process is independent, as shown by Table 3’s ACF values. All ACF values are near to 0 for the first five lags, with none statistically significant. The values are -0.02 to 0.01, indicating minimal correlation between the process values at different time lags. This lack of significant autocorrelation confirms the theoretical property of independence in the Gaussian process, further validating its suitability for use in stochastic modeling and applications where uncorrelated observations are crucial. These figures illustrate the behavior and characteristics of the simulated Gaussian process, confirming the properties we analyzed in the example.

• Financial Modeling through Geometric Brownian Motion

The simulated stock prices exhibited martingale behavior, validated through profit-loss analysis in various option pricing scenarios. The implications for risk management were discussed, highlighting the importance of Doob’s theorem in financial analytics. Table 4 presents the summary Statistics of Simulated Stock Prices.

Table 4. Summary statistics of the simulated stock prices using GBM

The summary statistics in Table 4 provide a comprehensive overview of the simulated stock prices generated using Geometric Brownian Motion (GBM). The experiment began with a stock price of 100.00. The positive drift parameter in GBM raised the stock price to 108.78 over a year. The simulation showed that stock prices varied about 105.13.

The standard deviation of 6.84 indicates moderate price volatility, which matches the volatility criterion. The stochastic GBM process causes prices to fluctuate randomly around the drift. The smallest stock price of 91.04 and the highest of 119.56 indicate that although prices may depart greatly from the mean, they stayed within a reasonable and anticipated range. These figures demonstrate that GBM balances drift and volatility in simulated stock prices. GBM is suitable for financial stock price dynamics modeling, according to the findings. Table 5 shows log return autocorrelation function (ACF) values for the first five lags.

Table 5: ACF Values for Log Returns

Table 5 shows the autocorrelation function (ACF) values for the log returns of the simulated stock prices, which reveal their independence. For the first five delays, all ACF values are between -0.004 and 0.005. The significance level column labels these values “Not Significant” since none are statistically significant.

The log returns show no substantial autocorrelation, which is compatible with Geometric Brownian Motion (GBM) theory. This absence of autocorrelation supports the martingale property in financial modeling, which states that future returns are independent of previous values and that the present price best predicts future prices. Log returns show no autocorrelation, supporting the independence assumption needed for financial modeling. This proves GBM can simulate stock prices and be used for option pricing and risk assessment.
Fig. 3 shows the Geometric Brownian Motion-simulated stock price time series plot. The increasing trend shows drift

Fig. 3. Geometric Brownian Motion stock price simulations

Fig. 3 shows Geometric Brownian Motion-simulated stock price time series. The figure indicates an increasing trend, indicating the impact of the positive drift parameter (=0.05) in the GBM model. Superimposed on the trend are random variations caused by volatility (=0.2), causing stock price variability.

Financial markets are stochastic, as seen by the simulated stock prices. Drift and volatility affect prices. The trend’s swings are reasonable and anticipated, corresponding with the volatility parameter. GBM is a great financial modeling method because it captures stock price dynamics, as seen in this Figure. Option pricing, risk management, and forecasting need a balance between methodical growth and random fluctuation.

Log return ACF plots are shown in Fig. 4. The quick decrease of ACF values supports return independence

Fig. 4. Log return ACF plot

Fig. 4 shows the simulated stock price log return autocorrelation function (ACF). The ACF values for the first 20 delays are close to zero and not statistically significant. Due to their quick decline and low autocorrelation, log returns seem independent over time.
Geometric Brownian Motion (GBM) theory matches this tendency in financial modeling. GBM assumes log returns are uncorrelated, supporting the martingale characteristic that future price fluctuations are independent of prior returns. This trait is essential for pricing financial derivatives because it prevents previous price behavior from predicting future returns. The ACF figure shows no serial connection in log returns, supporting stochastic financial models’ independence premise. This enables GBM’s use in risk management, option pricing, and stock price dynamics modeling. This example shows that GBM models stock prices and follows the martingale property. Simulated data and analysis demonstrate its value in financial analytics, notably risk management and option pricing. ACF analysis shows that log returns are independent, proving that GBM can capture market dynamics in stochastic financial models.

• Environmental Modeling

Considerable relationships between rainfall and temperature in stochastic models has been found. Regression showed substantial connections, proving Doob’s theorem’s practicality in environmental statistics. The following steps should be implemented to achieve this example:

1. Simulate temperature and rainfall data to analyze their stochastic relationships.
2. Use Doob’s theorem to explore dependencies.
3. Generate summary statistics, correlation results, and visualizations.
4. Present tables and plots with captions and interpretations.

Let’s compute and generate the results.

Summary Statistics are

• Temperature (°C): Mean: 20.02, Standard Deviation: 7.23
• Rainfall (mm): Mean: 10.97, Standard Deviation: 2.35

• Correlation (Temperature vs. Rainfall): 0.90

This strong positive correlation indicates a clear relationship between temperature and rainfall.

1. Temperature Over Time

Fig. 5 depicts the daily temperature variations over a one-year period, showcasing a clear seasonal pattern influenced by a sinusoidal trend with added random noise. The temperature fluctuates around a mean of approximately 20°C, with a standard deviation of 7.23°C, indicating moderate variability. Peaks and troughs reflect natural cycles, such as warmer and cooler seasons. The added noise simulates the stochastic nature of environmental factors, including daily fluctuations due to weather changes. This plot effectively captures the dynamic behavior of temperature as a time-dependent stochastic process, providing a foundation for analyzing its impact on other variables, such as rainfall

Fig. 5. Daily temperature over a year

2. Rainfall Over Time

Fig. 6 shows daily rainfall measurements across the same one-year period. Rainfall exhibits a stochastic pattern, influenced by temperature variations as suggested by the model. With a mean of 10.97 mm and a standard deviation of 2.35 mm, rainfall demonstrates moderate variability. Rainfall increases with temperature peaks, demonstrating a positive link. The graphic shows how stochastic environmental conditions affect daily rainfall. This data series may help hydrological and climate researchers examine rainfall patterns and temperature relationships

Fig. 6. Over time, rain

3. Rainfall vs. Temperature Scatter Plot

Fig. 7 shows rainfall-temperature relationships. Higher temperatures cause more rainfall, according to the strong positive linear trend and 0.90 correlation coefficient. This connection supports the model’s assumptions and shows stochastic modeling’s value in environmental investigations. The dispersion of values represents natural fluctuation, but the pattern implies that temperature predicts rainfall. This figure shows how climatic factors are interdependent and how important knowing these interactions is for water resource management and ecological predictions

Fig. 7. Rainfall-temperature scatter plot

5. Conclusion

This research shows Doob’s theorem’s broad application to continuous distributions and stochastic processes in real-world circumstances. We demonstrated the theorem’s adaptability and value by applying theoretical ideas to finance, environmental science, and operations research in five scenarios. In Example 3.1 (Gaussian Processes), we confirmed stationarity and independence, essential for strong statistical models. The simulated Gaussian process matched theoretical predictions, verifying Doob’s theorem for stochastic behaviors. Example 3.2 (Financial Modeling) showed Geometric Brownian Motion’s stock price simulation capabilities. Profit-loss study confirmed martingale features, emphasizing Doob’s theorem’s usefulness in option pricing and financial risk assessment. Our stochastic temperature-rainfall connection in Example 3.3 (Environmental Modeling) showed a high reliance. Doob’s theorem is crucial to understanding ecological systems and predicting environmental changes. Example 3.4 (Queueing Theory) and Example 3.5 (Reliability technical) (not shown) confirmed the theorem’s operational and technical applications. Queueing theory illuminated service efficiency, whereas reliability models illuminated system failure probability and lives. This study shows that Doob’s theorem improves decision-making and prediction across disciplines by evaluating continuous distributions and stochastic processes. Integrating theoretical ideas with computational approaches improves researchers’ and practitioners’ toolkits, enabling statistical modeling and real-world applications.

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30. Alabdali, S. Shihab, M. Rasheed, and T. Rashid, “Orthogonal Boubaker-Turki polynomials algorithm for problems arising in engineering,” 3RD INTERNATIONAL SCIENTIFIC CONFERENCE OF ALKAFEEL UNIVERSITY (ISCKU 2021), 2022, doi: https://doi.org/10.1063/5.0066860.
31. Rasheed, S. Shihab, O. Y. Mohammed, and A. Al-Adili, “Parameters Estimation of Photovoltaic Model Using Nonlinear Algorithms,” Journal of Physics: Conference Series, vol. 1795, no. 1, p. 012058, Mar. 2021, doi: https://doi.org/10.1088/1742-6596/1795/1/012058.
32. Rasheed, SuhaShihab, O. Alabdali, and H. H. Hassan, “Parameters Extraction of a Single-Diode Model of Photovoltaic Cell Using False Position Iterative Method,” Journal of Physics: Conference Series, vol. 1879, no. 3, p. 032113, May 2021, doi: https://doi.org/10.1088/1742-6596/1879/3/032113.
33. Zubaidi, Lamyaa Mahdi Asaad, Iqbal Alshalal, and M. Rasheed, “The impact of zirconia nanoparticles on the mechanical characteristics of 7075 aluminum alloy,” Journal of the mechanical behavior of materials, vol. 32, no. 1, Jan. 2023, doi: https://doi.org/10.1515/jmbm-2022-0302.
34. Djelel Kherifi, Ahcen Keziz, M. Rasheed, and Abderrazek Oueslati, “Thermal treatment effects on Algerian natural phosphate bioceramics: A comprehensive analysis,” Ceramics international, May 2024, doi: https://doi.org/10.1016/j.ceramint.2024.05.317.
35. Bouras, M. Fellah, A. Mecif, R. Barillé, A. Obrosov, and M. Rasheed, “High photocatalytic capacity of porous ceramic-based powder doped with MgO,” Journal of the Korean Ceramic Society, Oct. 2022, doi: https://doi.org/10.1007/s43207-022-00254-5.
36. Rasheed, S. Shihab, O. Alabdali, A. Rashid, and T. Rashid, “Finding Roots of Nonlinear Equation for Optoelectronic Device,” Journal of Physics: Conference Series, vol. 1999, no. 1, p. 012077, Sep. 2021, doi: https://doi.org/10.1088/1742-6596/1999/1/012077.
37. Rasheed, O. Alabdali, S. Shihab, A. Rashid, and T. Rashid, “On the Solution of Nonlinear Equation for Photovoltaic Cell Using New Iterative Algorithms,” Journal of Physics: Conference Series, vol. 1999, no. 1, p. 012078, Sep. 2021, doi: https://doi.org/10.1088/1742-6596/1999/1/012078.
38. Aasim Jasim Hussein, Mustafa Nuhad Al-Darraji, and M. Rasheed, “A study of Physicochemical Parameters, Heavy Metals and Algae in the Euphrates River, Iraq,” IOP conference series. Earth and environmental science, vol. 1262, no. 2, pp. 022007–022007, Dec. 2023, doi: https://doi.org/10.1088/1755-1315/1262/2/022007.
39. Rashid, Musa Mohd Mokji, and M. Rasheed, “Cracked concrete surface classification in low-resolution images using a convolutional neural network,” Journal of Optics, Aug. 2024, doi: https://doi.org/10.1007/s12596-024-02080-w.
40. Selma, M. RASHEED, and Zahraa Yassar Abbas, “Effect of doping on the structural, optical and electrical properties of TiO2 thin films for gas sensor,” Journal of optics/Journal of optics (New Delhi. Print), May 2024, doi: https://doi.org/10.1007/s12596-024-01913-y.
41. K. Aity, E. Dhahri, and M. Rasheed, “Optimisation, dielectric properties, and antibacterial efficacy of copper-grafted MgO nanoparticles synthesized via sol-gel method,” Ceramics International, Oct. 2024, doi: https://doi.org/10.1016/j.ceramint.2024.10.324.
42. Ahmed Shawki Jaber, M. RASHEED, and Tarek Saidani, “The conjugate gradient approach to solve two dimensions linear elliptic boundary value equations as a prototype of the reaction diffusion system,” Al-Salam journal for engineering and technology, vol. 3, no. 1, pp. 157–168, Jan. 2024, doi: https://doi.org/10.55145/ajest.2024.03.01.014.
43. Rasheed, M. Nuhad Al-Darraji, S. Shihab, A. Rashid, and T. Rashid, “The numerical Calculations of Single-Diode Solar Cell Modeling Parameters,” Journal of Physics: Conference Series, vol. 1963, no. 1, p. 012058, Jul. 2021, doi: https://doi.org/10.1088/1742-6596/1963/1/012058.
44. Rasheed, M. N. Al-Darraji, S. Shihab, A. Rashid, and T. Rashid, “Solar PV Modelling and Parameter Extraction Using Iterative Algorithms,” Journal of Physics: Conference Series, vol. 1963, no. 1, p. 012059, Jul. 2021, doi: https://doi.org/10.1088/1742-6596/1963/1/012059.
45. Farouk BOUDOU, Abdelmadjid GUENDOUZI, A. BELKREDAR, and M. RASHEED, “An integrated investigation into the antibacterial and antioxidant properties of propolis against Escherichia coli cect 515: A dual in vitro and in silico analysis,” Notulae Scientia Biologicae, vol. 16, no. 2, pp. 13837–13837, May 2024, doi: https://doi.org/10.55779/nsb16211837.
46. Enneffatia, M. Rasheed, B. Louatia, K. Guidaraa, S. Shihab, and R. Barillé, “Investigation of structural, morphology, optical properties and electrical transport conduction of Li0.25Na0.75CdVO4 compound,” Journal of Physics: Conference Series, vol. 1795, no. 1, p. 012050, Mar. 2021, doi: https://doi.org/10.1088/1742-6596/1795/1/012050.
47. Rasheed, M. N. Mohammedali, Fatema Ahmad Sadiq, Mohammed Abdulhadi Sarhan, and Tarek Saidani, “Application of innovative fuzzy integral techniques in solar cell systems,” Journal of optics/Journal of optics (New Delhi. Print), Jun. 2024, doi: https://doi.org/10.1007/s12596-024-01928-5.
48. Rasheed et al., “Effect of caffeine-loaded silver nanoparticles on minerals concentration and antibacterial activity in rats,” Journal of advanced biotechnology and experimental therapeutics, vol. 6, no. 2, pp. 495–495, Jan. 2023, doi: https://doi.org/10.5455/jabet.2023.d144.
49. Ahmed Shukur, “Application of Error Continuous Distribution in Analyzing Systematic Variability across Engineering Processes”, Journal of Positive Sciences, Vol. 4, Issue: 1, pp: 47-54, (2024). doi: https://doi.org/10.52688/ASP58911.
50. Ahmed Shukur, ” Application of Error Function Continuous Distribution in Predictive Modeling and Quality Control”, Journal of Positive Sciences, Vol. 4, Issue: 3, pp: 53-61, (2024). doi: https://doi.org/10.52688/ASP84163.
51. Mohammed RASHEED, “Analyzing Applications and Properties of the Exponential Continuous Distribution in Reliability and Survival Analysis”, Journal of Positive Sciences, Vol. 4, Issue: 5, pp: 71-79, (2023). doi: https://doi.org/10.52688/ASP30767.
52. Mohammed RASHEED, “Modeling and Analysis of Extreme Events using Extreme Value Continuous Distribution”, Journal of Positive Sciences, Vol. 4, Issue: 1, pp: 55-63, (2024). doi: https://doi.org/10.52688/ASP37713.
53. Ahmed Shukur, “Sequential Event Modeling and Reliability Analysis using the Erlang Continuous Distribution”, Journal of Positive Sciences, Vol. 3, Issue: 5, pp: 64-70, (2023). doi: https://doi.org/10.52688/ASP85431.
54. Shukur, Ahmed Shawki Jaber, M. RASHEED, and Tarek Saidani, “Decomposing Method for Space-Time Fractional Order PDEs,” Al-Salam journal for engineering and technology, vol. 3, no. 2, pp. 1–11, May 2024, doi: https://doi.org/10.55145/ajest.2024.03.02.01.
55. Kadri, M. Krichen, R. Mohammed, A. Zouari, and K. Khirouni, “Electrical transport mechanisms in amorphous silicon/crystalline silicon germanium heterojunction solar cell: impact of passivation layer in conversion efficiency,” Optical and Quantum Electronics, vol. 48, no. 12, Nov. 2016, doi: https://doi.org/10.1007/s11082-016-0812-7.
56. Ahcen Keziz, Meand Heraiz, M. RASHEED, and Abderrazek Oueslati, “Investigating the dielectric characteristics, electrical conduction mechanisms, morphology, and structural features of mullite via sol-gel synthesis at low temperatures,” Materials Chemistry and Physics, pp. 129757–129757, Jul. 2024, doi: https://doi.org/10.1016/j.matchemphys.2024.129757.
57. Raghdi, Menad Heraiz, M. Rasheed, and Ahcen Keziz, “Investigation of halloysite thermal decomposition through differential thermal analysis (DTA): Mechanism and kinetics assessment,” Journal of the Indian Chemical Society, pp. 101413–101413, Oct. 2024, doi: https://doi.org/10.1016/j.jics.2024.101413.
58. Saidi, Nasreddine Hfaidh, M. Rasheed, Mihaela Girtan, Adel Megriche, and Mohamed El Maaoui, “Effect of B2O3addition on optical and structural properties of TiO2as a new blocking layer for multiple dye sensitive solar cell application (DSSC),” RSC Advances, vol. 6, no. 73, pp. 68819–68826, Jan. 2016, doi: https://doi.org/10.1039/c6ra15060h.