Course detail

# Mathematics 3

FEKT-CMA3Acad. year: 2019/2020

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.

Supervisor

Department

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 CMA1 and CMA2 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.

Recommended or required reading

Fajmon, B., Růžičková, R.: Matematika 3. Elektronický text UMAT FEKT,
www.umat.feec.vutbr.cz/~fajmon/bma3, 2003. (CS)

Haluzíková, A.: Numerické metody. Skriptum FEI VUT Brno, 1989. (CS)

Zapletal, J.: Základy počtu pravděpodobnosti a matematické statistiky. Skriptum FEI VUT Brno, PC-DIR 1995. (CS)

Ralston, A.: Základy numerické matematiky. Praha, Academia 1978. (CS)

Vitásek, E.: Numerické metody. Praha, SNTL 1987. (CS)

Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, Alfred A. Knopf, New York 1988. (EN)

Taha, H.A.: Operations Research. An Introduction. Fourth Edition, Macmillan Publishing Company, New York 1989. (EN)

NOVÁK, M. Matematika 3: Sbírka úloh z numerických metod. Brno: FEKT VUT, 2010. (CS)

HLAVIČKOVÁ, I.; HLINĚNÁ, D. Matematika 3 - Sbírka úloh z pravděpodobnosti. Matematika 3 - Sbírka úloh z pravděpodobnosti. Brno: UMAT FEKT VUT, 2007. s. 1-77. (CS)

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Written examination is evaluated by maximum 70 points, the student's work during the semestr is assesed by maximum 30 points (2 test up to 15 points each).

Language of instruction

English

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.

#### Type of course unit

eLearning

**eLearning:** currently opened course