Faniya Ahmetova

In work the method of presentation of the topic "Functions of random variables" in the course "Probability theory". The article is written based on years of experience teaching this subject and will be useful for both students and teachers in conducting practical classes. There are no proofs of the theorems used, however, a list of references that can be consulted for more detailed explanations. Considered a large number of examples that will allow students to learn the material under study to the extent necessary. The aim of this work is to help students to acquire skills in the application of probability methods to the solution of various problems.

The paper proposes a technique for presenting the topic "The formula of full probability and the Bayes formula" based on the personal experience of the authors in the discipline "Theory of Probability". When solving problems on this topic, the greatest difficulty for students is calculation of conditional probabilities. In this connection, special attention is paid to the method of solving problems. The main theoretical information and a large number of typical examples showing solutions that allow students to acquire the necessary skills in mastering this topic are given.

The paper considers a brief theoretical information related to the application of operational calculus. The authors tabulate originals and images, basic theorems and models for the location of the image for the original; describe the ways to restore the original according to the known image. Since the classical methods of solutions of the Cauchy problem for linear differential equations with constant coefficients and right-hand side in the form of a composite function are highly intensive, clearly shows the efficiency of application of operational calculus and solutions of such problems. The parsed solution of the Cauchy problem for the Duhamel formula.

The paper considers the method of bringing the equations of curves and surfaces of the second order to the canonical form. The stages of practical calculating the orthogonal transformation resulting in a quadratic form to canonical form are illustrated on the examples. The prospects of using the application package in the educational process (namely, MathCAD environment) are shown. The procedure of finding the eigenval-ues and eigenvectors with this tool is demonstrated on the examples. Their finding is usually time-consuming, so for the speed of counting, it is advisable to use MathCAD program. The paper will be useful to students and lecturers during seminars on the subject "Linear Algebra".

The paper considers the Fourier transform of impulse functions, which is the mathematical basis of the tasks associated with the theory of reception and signal conversion in optoelectronic system. The method of calculating of two functions convolution, its Fourier image and the image of the Fourier-Bessel axisymmetric functions are demonstrated in details. A table, which summarizes the analytical expression for the shifted impulse functions and records their Fourier transforms, is provided. A wide range of examples of solving tasks, containing the graphic illustration, is analyzed. A structured approach to the presentation of the material, which combines basic theoretical information and analysis of typical tasks, will help second-year students of optoelectronic specialty in their independent work and homework.