Course detail

# Mathematics 3

The aim of this course is to introduce the basics of two mathematical disciplines: numerical methods, and probability and statistics.
In the field of probability, main attention is paid to random variables, both discrete and continuous. The end of the course of probability is devoted to hypothesis testing.
In the field of numerical mathematics, the following topics are covered: root finding, systems of linear equations, curve fitting (interpolation and splines, least squares method), numerical differentiation and integration.

Learning outcomes of the course unit

Students completing this course should be able to:
In the field of probability and statistics:
- compute the basic characteristics of statistical data (mean, median, modus, variance, standard deviation)
- choose the correct probability model (classical, discrete, geometrical probability) for a given problem and compute the probability of a given event
- compute the conditional probability of a random event A given an event B
- recognize and use the independence of random events when computing probabilities
- apply the total probability rule and the Bayes' theorem
- work with the cumulative distribution function, the probability mass function of a discrete random variable and the probability density function of a continuous random variable
- construct the probability mass functions (in simple cases)
- choose the appropriate type of probability distribution in model cases (binomial, hypergeometric, exponential, etc.) and work with this distribution
- compute mean, variance and standard deviation of a random variable and explain the meaning of these characteristics
- perform computations with a normally distributed random variable X: find probability that X is in a given range or find the quantile/s for a given probability
- construct estimates of uknown parameters of the known distribution
- estimate parameters of a probability distribution by means of the maximum likelihood method.
In the field of numerical methods, the student should be able to:
- find the root of a given equation f(x)=0 using the bisection method, Newton method or the iterative method, describe these methods including the convergence conditions
- find the root of a system of two equations using Newton or iterative method
- solve a system of linear equations using Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iteration methods, discuss the advantages and disadvantages of these methods
- find Lagrange or Newton interpolation polynomial for given points and use it for approximating the given function
- find the approximation of a function by spline functions
- find the approximation of a function given by table of points by the least squares method (linear, quadratic or exponential approximation)
- choose the most convenient type of approximation (interpolation polynomial, spline, least squares)
- estimate the derivative of a given function using numerical differentiation
- compute the numerical approximation of a definite integral using trapezoidal and Simpson method, describe the principal of these methods, compare them according to their accuracy
- find the approximate solution of a differential equation using Euler method, modified Euler methods and Runge-Kutta methods

Prerequisites

The student should be able to apply the basic knowledge of combinatorics on the secondary school level: to explain the notions of variations, permutations and combinations, to determine their counts, to perform computations with factorials and binomial coefficients. From the BMA1 and BMA2 courses, the basic knowledge of differential and integral calculus is demanded. Especially, the student should be able to sketch the graphs of elementary functions, to substitute into functions, to compute derivatives (including partial derivatives) and integrals.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

FAJMON, B., HLAVIČKOVÁ, I., NOVÁK, M. Matematika 3. Elektronický text FEKT VUT, Brno, 2013 (CS)
Fajmon, B., Hlavičková, I., Novák, M. Matematika 3. Elektronický text FEKT VUT, Brno, 2013 (CS)

Planned learning activities and teaching methods

Teaching methods include lectures, computer exercise and other activities.

Assesment methods and criteria linked to learning outcomes

The student's work during the semestr (written tests and projects) is assessed by maximum 30 points.
Written examination is evaluated by maximum 70 points. It consist of several tasks and two theoretical questions (1+1, each for 5 points). To pass the exam, the student must gain at least 10 points in probability and at least 10 points in numerical methods.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem.
2. Random variables, random vector, distribution function.
3. Characteristics of random variables, basic distributions.
4. Characteristics of random vectors, covariance, correlation.
5. Law of large numbers, central limit theorem.
6. Introduction to statistics. histogram, point and interval estimates of the mean and variance.
7.Moment methods, maximum likelihood method.
8. Hypothesis testing about the mean and variance, goodness -of-fit tests.
9. Correlation tests, non-parametric tests.
10. Introduction to numerical methods. Numerical methods for root finding (bisection method, Newton method, iterative method)
11. Numerical solution of systems of nonlinear equations. Systems of linear equations (Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iterative methods).
12. Interpolation: interpolation polynomial (Lagrange and Newton), splines (linear and cubic)
13. Least squares approximation. Numerical differentiation and integration.

Aims

The aim of this course is to introduce the basics of two mathematical disciplines: numerical methods, and probability and statistics.

Specification of controlled education, way of implementation and compensation for absences

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Classification of course in study plans

• Programme BPC-AUD Bachelor's

specialization AUDB-ZVUK , 2. year of study, winter semester, 5 credits, compulsory
specialization AUDB-TECH , 2. year of study, winter semester, 5 credits, compulsory

• Programme BPC-EKT Bachelor's, 2. year of study, winter semester, 5 credits, compulsory
• Programme BPC-MET Bachelor's, 2. year of study, winter semester, 5 credits, compulsory
• Programme BPC-TLI Bachelor's, 2. year of study, winter semester, 5 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Fundamentals seminar

4 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

18 hours, compulsory

Teacher / Lecturer

The other activities

4 hours, compulsory

Teacher / Lecturer

eLearning