تحليل رياضي لتوزيع بيتا وتطبيقاته في معالجة الإشارات الكهربائيةالعشوائية

Abstract

function and the Beta distribution, both of which play a crucial r ed ole in applimathematics and statistical modeling. The Beta distribution is directly derived from the Beta function and is widely used in statistical applications that require modeling probabilities within a bounded interval. In electrical engineering, probability distributions are essential tools for analyzing random systems and signal processing, particularly in addressing challenges related to noise, signal quality, and statistical estimation. The study focuses on the mathematical relationship between the Beta function and the Beta distribution, highlighting their relevance in analyzing stochastic electrical signals. By leveraging these functions, engineers can model uncertainty, optimize signal performance, and enhance the reliability of communication systems. The research also presents practical applications, including the design of probabilistic filters, estimation of transmission error probabilities, and performance enhancement in communication networks Findings indicate that the Beta distribution offers high flexibility in modeling signal behavior under varying noise conditions, making it a valuable tool for improving statistical accuracy in signal analysis. Its adaptability allows for more precise error

estimation and better filter design, which are critical in modern communication systems where data integrity and transmission efficiency are paramount Moreover, the integration of Beta-based models into signal processing frameworks demonstrates a promising approach to bridging theoretical mathematics with real- world engineering challenges. This synergy not only improves system performance but also opens new avenues for developing intelligent, adaptive technologies capable of responding to dynamic environments. In conclusion, the Beta function and distribution serve as powerful analytical tools in electrical engineering, offering robust solutions for signal modeling, error prediction, and system optimization in stochastic settings