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# Probability Theory

2020/2021
ENG
Instruction in English
6
ECTS credits
Course type:
Compulsory course
When:
1 year, 1, 2 module

### Course Syllabus

#### Abstract

The objective of the discipline "Probability Theory" is to lay the foundation of probability for all courses that follow in the “Master of Applied Statistics with Network Analysis” program. The course is strongly related and complementary to other compulsory courses provided in the first year (e.g. Applied Linear Models II, Contemporary Data Analysis) and sets a crucial prerequisite for later courses and research projects as well as for the master thesis.

#### Learning Objectives

• The course gives students an important foundation to develop and conduct their own research as well as to evalu- ate research of others.

#### Expected Learning Outcomes

• Know the role of probability theory in the sciences, communicate the ideas and results of probability.
• Know joint probability distributions, expectation, variance and covariance of random variables.
• Be able to define and apply the concepts of sample space, events, probability, random variables, and their distributions.
• Have an understanding of the basic principles of probability and lay the foundation for future learning in the area.
• Be able to formulate and apply theorems concerning functions of random variables and the moment- generating functions, Chebyshev’s theorem, the Central Limit Theorem and the Law of Large Numbers.
• Be able to formulate and apply the definitions of convergence in distribution and in probability, formulate scientific problems involving randomness in mathematical terms.
• Know the basic principles of using probability for using analytic models.
• Be able to use probability in future courses and analytical career overall.
• Have the skill to meaningfully develop an appropriate model for the research question, using probability theory.
• Have the skill to work with statistical software, required to analyze the data.

#### Course Contents

• Axioms of probability
The first session will cover several topics: sample space and events. Axioms of probability. Clas-sical probability: finite sample spaces having equally likely outcomes. Probability as a continuous event function. Probability as a measure of belief.
• Conditional probability and independence
Conditional probabilities. The multiplication rule. Formula of total probability. Bayes’s formula. Independent events. Bernstein example.
• Discrete Random Variables
Random variables. Discrete random variables. Expected value. Expectation of a function of a random variable. The Bernoulli and binomial random variables. The Poisson random variable. Other discrete probability distributions. Expected value of sums of random variables. Properties of the cumulative distribution function. Generation functions for positive integer-valued random variables. Branching processes. The probability of ultimate extinction for branching processes.
• Continuous Random Variables
Probability density function. Expectation and variance of continuous random variables. The uniform random variable. Normal random variables. Exponential random variables. The continuous distributions. The distribution of a function of a random variable.
• Jointly Distributed Random Variables
Joint distribution functions. Independent random variables. Sums of independent random varia-bles. Conditional distributions: discrete case and continuous case. Joint probability distribution of functions of random variables.
• Properties of Expectation
Covariance, variance of sums. Correlations. Conditional expectation. Conditional expectation and prediction. The continuous Bayes' rule.
• Limit Theorems
Chebyshev’s inequality and the weak law of large numbers. The central limit theorem. The strong law of large numbers. Monte–Carlo method. Other inequalities. Bounding the error probability when approximating a sum of independent Bernoulli random variables by a Poisson random variable.
(will span over several sessions) Discrete-Time Markov Chains. Classification of States. Steady-State Behavior. Uncertainty and entropy.

#### Assessment Elements

• Final In-Class and/or Take-home exam (at the discretion of the instructor)
• Class participation
• 3 home assignments, each assignments x 20% each

#### Interim Assessment

• Interim assessment (2 module)
0.6 * 3 home assignments, each assignments x 20% each + 0.1 * Class participation + 0.3 * Final In-Class and/or Take-home exam (at the discretion of the instructor)

#### Recommended Core Bibliography

• Rohatgi, V. K., & Saleh, A. K. M. E. (2001). An Introduction to Probability and Statistics (Vol. 2nd ed. Vijay K. Rohatgi, A.K. Md. Ehsanes Saleh). New York: Wiley-Interscience. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=396326
• Venkatesh, S. S. (2013). The Theory of Probability : Explorations and Applications. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=498312