714-0268/01 – Mathematics III (BcM3)

The aim of the course is to provide theoretical and practical foundation for understanding of the meaning of basic probability terms and teach the student to statistical thinking as a way of understanding of the processes and events around us, to acquaint him with the basic methods of statistical data gathering and analyzing, and to show how to use these general procedures in other subjects of study and in practice. Graduates of this course should be able to: • understand and use the basic terms of combinatorics and probability theory; • formulate questions that can be answered by the data, learn the principles of data collecting, processing and presenting; • select and use appropriate statistical methods for data analysis; • propose and evaluate conclusions (inferences) and predictions using the data.

Combinatorics and probability. Random events, operations with them, sample space. Definitions of events' probability - classical, geometrical, statistics. Conditional probability. Total probability and independent events. Random variable and its characteristics. Basic types of probability distributions of discrete random variables. Basic types of probability distributions of continuous random variables. Random vector, probability distribution, numerical characteristics. Statistical file with one factor. Grouped frequency distribution. Statistical file with two factors. Regression and correlation. Random sample, point and interval estimations of parameters. Hypothesis testing.iables: two-dimensional integrals, three-dimensional integrals, line integral of the first and the second kind. Probabilities of random events: axioms of probability, conditional probability, independence. Random variables: discrete random variables, continuous random variables, expected values. Important practical distributions of discrete and continuous random variables.

Kučera, Radek: Mathematics III, VŠB – TUO, Ostrava 2005, ISBN 80-248-0802-1

Passing the course, requirements Course-credit -participation on tutorials is obligatory, 20% of absence can be apologized, -elaborate programs, -pass the written tests, Point classification: 5-20 points. Exam Practical part of an exam is classified by 0 - 60 points. Practical part is successful if student obtains at least 25 points. Theoretical part of the exam is classified by 0 - 20 points. Theoretical part is successful if student obtains at least 5 points. Point quantification in the interval 100 - 86 85 - 66 65 - 51 50 - 0 National grading scheme excellent very good satisfactory failed 1 2 3 4 List of theoretical questions: 1. Definition of Double Integral. 2. Double Integrals over a Rectangular Region. 3. Vertically Simple Region. 4. Horizontally Simple Region. 5. Double Integrals over General Region,. Fubini’s Theorem. 6. Double Integrals in Polar Coordinates. 7. Jacobian, Jacobian for Polar Coordinates. 8. Generalized Polar Coordinates. 9. Double Integrals as Volume or Area. 10. Area of Region. 11. Centre of Gravity of a two-dimensional Object.. 12. Second Moment of Area. 13. Definition of the Triple Integral. 14. Dirichlet’s Theorem for Evaluating Triple Integrals. 15. Normal domains on R3 19. Triple Integrals over Regular Regions, Fubini’s Theorem. 20. Cylindrical Coordinates. 21. Spherical Coordinates. 22. Jacobian for Cylindrical Coordinates. 23. Jacobian for Spherical Coordinates. 24. Volume of the three-dimensional Region. 25. First Moments about the Coordinate Planes. 26. Mass Moment of Inertia. 27. Center of Mass of the Body. 28. Curve in R3. 29. Line Integral of a Scalar Field. Evaluating of Line Integrals. 30. Geometrical Acceptation. 31. Line Integral of a Vector Field. 32. Finding the Work done on an Object in a Force Field. 33. Line Integrals; Green's Theorem. 34. Application of Green’s Theorem, Area of a Region 35. Outcomes and Events. 36. Classical Definition of Probability. 37. Statistical Definition of Probability. 38. Probability Sum of Random Variables. 39. Conditional Probability. 40. Product rule. Independence. 41. Bernoulli Process, Independent Random Variables. 42. Random Variables. 43. Discrete Probability Distributions. 44. Continuous Probability Distributions. 45. Cumulative Distribution Function, Properties. 46. Graphical Display of Discrete Probability Distributions. 47. Binomial Distribution. 48. Uniform distribution. 49. Probability Frequency Function. 50. Properties of Probability Frequency Functions. 51. Normal distribution. 52. Standard (i.e., mean=0, standard deviation=1) Normal Distribution. 53. Transformation to Standard Normal Distribution. 54. rth Moment of Random Variable. 55. rth Central Moment. 56. Standard Moments. 57. Characteristics of location and of variability. 58. Obliqueness and Acuteness Characteristics. 59. Introduction to statistics. 60. Selection Moments.