Probability Theory

Learning Outcomes

The course’s material includes mathematical definitions, fundamental concepts, results and reasoning methodologies that pertain to basic Probability Theory objects and models underlying the foundations and applications of computer science. Moreover the relevant connections of probability theory to several more specific branches of computer science are presented. The course directly supports most of the curriculum subjects and lessons: Note that during the lesson, specific examples of application are discussed in some of the above curricular subjects such as applications in telecommunication systems, cryptography, digital services such as e- learning, e-health using new technologies with the help of programs such as Matlab, Octave, SPSS and R.

Upon successfully completion of the course the students will be in position to:·

• Know, describe the basic knowledge of Probability Theory (indicatively: probability definition, combinatorial analysis, random variables, discrete and continuous distributions, two-dimensional probabilistic generators and generators, etc.) basic values and the basic tools (probability mass and density functions, etc.) of Probability Theory.·
• Calculate the probabilities of possible non-trivial probability problems using probability theory, combinatorial and calculus.·
• Identify and know well-known probabilistic models in Computer Science problems.·
• Choose the appropriate mathematical method so that it can model the problem that it is called upon to solve and compile new probabilistic models for problems and systems that appear in computer science using simpler probabilistic models.·
• Know, handle, and understand Matlab, Octave, SPSS, and R programs on probability theory applications in computer science.

Course Contents

• Accidental experiment, samples and possibilities.
• Definitions of possibilities.
• Finite samples with results of equal possibilities.
• Provisions, combinations, binomial theorem.
• Committed probability.
• The multiplicative theorem.
• Total probability and Bayes theorem.
• Independent trials.
• Random variables, probability distributions.
• Parameters of distributions, interrelation of distribution accidental variables.
• One-dimensional distributions.
• Continuous distributions.
• Discrete distributions
• Generators of proneness; probabilities generators.
• Two-dimensional distributions.
• Marginal distributions distributions, Average-Committed distributions
• Transformations of random variables-Stochastic independence of two random variables.

Recommended Readings

• M. Filippakis, Probability Theory and elements of statistics analysis, applications with python, matlab, R, Spss, Tsotras Publications, Athens 2019, 1rst Edition.
• M. Filippakis-T. Papadoggonas, Probability Theory and Business Statistics, Tsotras Publications, Athens 2017, 1rst Edition.
• M. Koutras, Probability Theory and applications, tsotras publications, Athens 2015.
• Durrett R. (2004): Probability: Theory and Examples, 3rd Edition, Duxbury Press.
• Teaching Notes.