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, numerical solving of differential equations.

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
- approximate the binomial distribution with help of the normal distribution
- perform simple hypothesis testing: Z-test, test on the mean of normal distribution variance known, test on the parameter p of the binomial distribution

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., RŮŽIČKOVÁ, I. MATEMATIKA_3_S. PDF. Matematika 3. Brno: UMAT FEKT VUT, 2003. s. 1 ( s.) (CS)
Haluzíková, A. Numerické metody. Skriptum FEI VUT. Brno: VUT, 1989. (CS)
Zapletal, J. Základy počtu pravděpodobnosti a matematické statistiky. Skriptum FEI VUT. Brno: PC-DIR, 1995. (CS)
Vitásek, E. Numerické metody. Praha: SNTL, 1987. (CS)
Loftus, J., Loftus, E. Essence of Statistics. Second Edition, New York: Alfred A. Knopf, 1988. (EN)
Taha, H.A. Operations Research. An Introduction. Fourth Edition, New York: Macmillan Publishing Company, 1989. (EN)
Hlavičková, I., Hliněná, D.: Matematika 3 - sbírka úloh z pravděpodobnosti. Elektronický text FEKT VUT, Brno, 2010 (CS)
Novák, M.: Matematika 3 - sbírka příkladů z numerických metod. Elektronický text FEKT VUT, Brno, 2010 (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 homework) is assessed by maximum 30 points.
Written examination is evaluated by maximum 70 points. It consist of four tasks (two in probability and two in numerical methods, each for 15 points) 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 descriptive statistics
2. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem.
3. Discrete random variables (probability mass function, cumulative distribution function, mean and variance).
4. Discrete probability distributions (binomial, geometric, hypergeometric, Poisson).
5. Continuous random variables (probability density function, distrubution function, mean, variance, quantiles). Exponencial distribution.
6. Normal distribution. Central limit theorem. Normal approximation to the binomial distribution.
7. Introduction to statistics. Z-test. Test of the mean of a normal distrinution, variance known.
8. Introduction to numerical methods. Numerical methods for root finding (bisection method, Newton method, iterative method)
9. Numerical solution of systems of nonlinear equations. Systems of linear equations (Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iterative methods).
10. Interpolation: interpolation polynomial (Lagrange and Newton), splines (linear and cubic)
11. Least squares approximation. Numerical differentiation.
12. Numerical integration (trapezoidal and Simpson method).
13. Numerical solution of differential equations: initial problems (Euler method and its modifications, Runge-Kutta methods), boundary value problems (very briefly).

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 EEKR-B Bachelor's

branch B-AMT , 2. year of study, winter semester, 5 credits, compulsory
branch B-EST , 2. year of study, winter semester, 5 credits, compulsory
branch B-MET , 2. year of study, winter semester, 5 credits, compulsory
branch B-SEE , 2. year of study, winter semester, 5 credits, compulsory
branch B-TLI , 2. year of study, winter semester, 5 credits, compulsory

• Programme EEKR-CZV lifelong learning

branch ET-CZV , 1. year of study, winter semester, 5 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Exercise in computer lab

14 hours, compulsory

Teacher / Lecturer

The other activities

12 hours, compulsory

Teacher / Lecturer