### Introduction to Probability (MATH 203)

2023 Fall
Faculty of Engineering and Natural Sciences
Mathematics(MATH)
3
6
Kağan Kurşungöz kursungoz@sabanciuniv.edu, Gökalp Alpan gokalp.alpan@sabanciuniv.edu,
English
MATH102
Formal lecture,Recitation

### CONTENT

Counting techniques, combinatorial methods, random experiments, sample spaces, events, probability axioms, some rules of probability, conditional probability, independence, Bayes' theorem, random variables (r.v.'s), probability distributions, discrete and continuous r.v.'s, probability density functions, multivariate distributions, marginal and conditional distributions, expected values, moments, conditional expectation, Chebyshev's theorem, product moments, moments of linear combinations of r.v.'s, special discrete distributions, uniform, Bernoulli, binomial, negative binomial, geometric, hypergeometric and Poisson distributions, special probability densities, uniform, gamma, exponential and normal densities, normal approximation to binomial, distribution of functions of r.v.'s, distribution function and moment-generating law of large numbers, the central limit theorem, function techniques, distribution of the mean, basic methods for statistical estimation and hypothesis testing.

### OBJECTIVE

- To give an understanding of uncertainty and randomness
- To teach the fundemantal concepts and defintions of probability
- To equip the students with the tools and techniques of probability theory

### LEARNING OUTCOMES

• Use the basic principles of counting, permutations, combinations, and multinomial coefficients.
• Perform set operations and compute elementary (conditional) probabilities.
• Use the concept of random variables and their distributions, cumulative distribution functions.
• Compute marginal distributions, conditional distributions and conditional expectations.
• Evaluate mathematical expectations, moments, variances, co-variances, conditional expectations, and moment-generating functions.
• Investigate the basic properties of important discrete and continuous random variables such as Bernouilli, binomial, hypergeometric, Poisson, uniform, exponential, gamma and normal.
• Implement different techniques such as the distribution function technique, the moment generating function technique to evaluate the distribution of functions of random variables.
• Investigate the properties of various statistics (eg. sample mean, sample variance, order statistics) and use them in simple statistical estimation and hypothesis testing problems.
• Applications in various disciplines.

### PROGRAMME OUTCOMES

1. Understand the world, their country, their society, as well as themselves and have awareness of ethical problems, social rights, values and responsibility to the self and to others. 2

2. Understand different disciplines from natural and social sciences to mathematics and art, and develop interdisciplinary approaches in thinking and practice. 4

3. Think critically, follow innovations and developments in science and technology, demonstrate personal and organizational entrepreneurship and engage in life-long learning in various subjects; have the ability to continue to educate him/herself. 4

4. Communicate effectively in Turkish and English by oral, written, graphical and technological means. 3

5. Take individual and team responsibility, function effectively and respectively as an individual and a member or a leader of a team; and have the skills to work effectively in multi-disciplinary teams. 3

1. Possess sufficient knowledge of mathematics, science and program-specific engineering topics; use theoretical and applied knowledge of these areas in complex engineering problems. 4

2. Identify, define, formulate and solve complex engineering problems; choose and apply suitable analysis and modeling methods for this purpose. 3

3. Develop, choose and use modern techniques and tools that are needed for analysis and solution of complex problems faced in engineering applications; possess knowledge of standards used in engineering applications; use information technologies effectively. 2

4. Have the ability to design a complex system, process, instrument or a product under realistic constraints and conditions, with the goal of fulfilling specified needs; apply modern design techniques for this purpose. 2

5. Design and conduct experiments, collect data, analyze and interpret the results to investigate complex engineering problems or program-specific research areas. 2

6. Possess knowledge of business practices such as project management, risk management and change management; awareness on innovation; knowledge of sustainable development. 1

7. Possess knowledge of impact of engineering solutions in a global, economic, environmental, health and societal context; knowledge of contemporary issues; awareness on legal outcomes of engineering solutions; knowledge of behavior according to ethical principles, understanding of professional and ethical responsibility. 1

8. Have the ability to write effective reports and comprehend written reports, prepare design and production reports, make effective presentations, and give and receive clear and intelligible instructions. 2

1. Develop knowledge of theories, concepts, and research methods in humanities and social sciences. 1

2. Assess how global, national and regional developments affect society. 1

3. Know how to access and evaluate data from various sources of information. 1

1. Comprehend key concepts in biology and physiology, with emphasis on molecular genetics, biochemistry and molecular and cell biology as well as advanced mathematics and statistics. 1

2. Develop conceptual background for interfacing of biology with engineering for a professional awareness of contemporary biological research questions and the experimental and theoretical methods used to address them. 1

1. Design, implement, test, and evaluate a computer system, component, or algorithm to meet desired needs and to solve a computational problem. 1

2. Demonstrate knowledge of discrete mathematics and data structures. 1

3. Demonstrate knowledge of probability and statistics, including applications appropriate to computer science and engineering. 4

1. Use mathematics (including derivative and integral calculations, probability and statistics, differential equations, linear algebra, complex variables and discrete mathematics), basic sciences, computer and programming, and electronics engineering knowledge to (a) Design and analyze complex electronic circuits, instruments, software and electronics systems with hardware/software or (b) Design and analyze communication networks and systems, signal processing algorithms or software 4

1. Applying fundamental and advanced knowledge of natural sciences as well as engineering principles to develop and design new materials and establish the relation between internal structure and physical properties using experimental, computational and theoretical tools. 1

2. Merging the existing knowledge on physical properties, design limits and fabrication methods in materials selection for a particular application or to resolve material performance related problems. 1

3. Predicting and understanding the behavior of a material under use in a specific environment knowing the internal structure or vice versa. 1

1. Familiarity with concepts in statistics and optimization, knowledge in basic differential and integral calculus, linear algebra, differential equations, complex variables, multi-variable calculus, as well as physics and computer science, and ability to use this knowledge in modeling, design and analysis of complex dynamical systems containing hardware and software components. 2

2. Ability to work in design, implementation and integration of engineering applications, such as electronic, mechanical, electromechanical, control and computer systems that contain software and hardware components, including sensors, actuators and controllers. 1

1. Formulate and analyze problems in complex manufacturing and service systems by comprehending and applying the basic tools of industrial engineering such as modeling and optimization, stochastics, statistics. 2

2. Design and develop appropriate analytical solution strategies for problems in integrated production and service systems involving human capital, materials, information, equipment, and energy. 2

3. Implement solution strategies on a computer platform for decision-support purposes by employing effective computational and experimental tools. 1

1. Provide constructive analysis of economic phenomena at the national and international level, and interactions between the two. 1

2. Develop an understanding of organizations and institutions in the society as well as their influence on the economy. 1

3. Recognize how incentives shape the behavior of individuals and organizations. 1

4. Identify "economic" problems and propose alternative models and/or design and conduct research to provide viable solutions using theoretical tools and/or quantitative methods. 1

5. Communicate problems and solutions to managerial and policy decision-making units as well as to lay audiences. 1

1. Analyze global affairs from international relations and economics perspectives. 1

2. Demonstrate theoretical and practical knowledge of the international affairs. 1

3. Compete for increasing opportunities in careers within the newly emerging global institutions. 1

4. Evaluate the international political events and present their views and positions on international affairs with advanced oral and written skills. 1

1. Understand and follow changes in patterns of political behavior, ideas and structures. 1

2. Develop the ability to make logical inferences about social and political issues on the basis of comparative and historical knowledge. 1

### ASSESSMENT METHODS and CRITERIA

 Percentage (%) Final 40 Midterm 60 Quiz 5