METHODS OF USING ELEMENTS OF PROBABILITY THEORY IN PROVING INEQUALITIES (Scientific journal of the Fergana State University, 2024)
The article gives a method for proving certain types of inequalities using elements of probability theory. It is known that inequalities are a tool used in evaluating and justifying many mathematical conclusions. The methods of their proof lead to the integration of many mathematical fields and provide the researcher with new knowledge. At the same time, proving any inequality requires researchers to have special knowledge and methodology. This makes it necessary to develop the most effective ways to prove some typical inequalities. For this purpose, a proof method was developed using elements of probability theory for inequalities of a certain type. This methodology is based on the law of distribution of random variables and the adoption of non-negative variance values. Using this property of dispersion, some relations for inequalities are given and proven. The fact that the inequalities given by this relation are easier to prove than other methods has been proven by solving a number of practical problems. In general form, this statement can be used to prove some complex inequalities. This means that knowledge of probability theory plays an important role in other areas of mathematics.