NUMERICAL CHARACTERISTICS OF RANDOM VARIABLES
Keywords:
random variable, numerical characteristics, mathematical expectation, variance, standard deviation, skewness coefficient, kurtosis, moments, quantile, median, empirical distribution, interval estimation, probability theory, statistical analysis, financial risk assessment, quality control.Abstract
Numerical characteristics of random variables are one of the fundamental branches of mathematical statistics and probability theory, allowing for the quantitative description and analysis of the properties of random events. This article provides an in-depth discussion of the theoretical foundations, mathematical expressions, and practical significance of the main numerical characteristics of random variables—such as the mean, variance, standard deviation, skewness coefficient, kurtosis, moments, quantiles, and median.
The numerical characteristics of a random variable help to fully describe its distribution pattern. The mathematical expectation shows the average value of a random variable, the variance and standard deviation express the degree of its spread, and the skewness and kurtosis express the shape and deviation of the distribution. These characteristics are widely used in probability theory, statistical analysis, financial mathematics, quality control, insurance business, and many other scientific and practical areas.
The article considers numerical characteristics for discrete and continuous types of random variables separately. The exact formulas for the mathematical expectation and variance for discrete random variables, and the integral expressions and their properties for continuous random variables are analyzed in detail. The relationship between empirical and theoretical characteristics, methods for their estimation (point and interval estimates), estimation methods such as the method of moments and the maximum likelihood method are also covered.
The article shows with examples how the numerical characteristics of random variables are used in modern scientific research, their role in financial risk assessment, quality control, medical statistics, and machine learning algorithms. In particular, the importance of these characteristics in statistical data analysis in the conditions of Uzbekistan and current issues in their application in practice are also discussed.
This study serves as a valuable scientific resource for students, scientists, and professionals in mathematics, statistics, and applied sciences by providing a deep theoretical and practical understanding of the topic of numerical characteristics of random variables. The article helps to better understand the properties of random variables and their effective application in various fields.
