To make students understand the basic concepts of the probability theory: Sample Spaces, Events, Discrete and Continuous sample, Kolmogorov's axioms, the conditional probability of events, independence and Bayes' formula of total probability, Bernoulli scheme, random variables and distribution of random variables.

Learning Outcomes

CTPO

TOA

Upon successful completion of the course, the students will be able to :

LO - 1 :

understand the basic concepts of probability theory

2,8

1,

LO - 2 :

construct a mathematical model of stochastic experiment

2,8

1,

LO - 3 :

have the ability to calculate probability of events

2,8

1,

CTPO : Contribution to programme outcomes, TOA :Type of assessment (1: written exam, 2: Oral exam, 3: Homework assignment, 4: Laboratory exercise/exam, 5: Seminar / presentation, 6: Term paper), LO : Learning Outcome

Contents of the Course

Historical development and subject of the probability theory. Sample space. Events and operations over events. Frequency of event. Definition of the probability in discrete sample space. Classical probability. Algebra and Borel sigma algebra. Definition of the probability in continuous sample space. Kolmogorov's axioms. Probability space. Properties of the probability measure. Geometrical probability. Independence of events. Formula of product of probabilities. Conditional probability. Total probability formula. Bayes's formulas. Sequence of independent trails. Bernoulli scheme. Limit theorems in Bernoulli scheme (Mouavr-Laplace's local and integral formulas, Poisson's theorem, Law of large numbers). Polynomial scheme. Sample Markov chains. Concept of random variable. Operations over random variables. Distribution function and its properties.

Course Syllabus

Week

Subject

Related Notes / Files

Week 1

Historical development and subject of the probability theory.

Week 2

Sample space. Events and operations over events.

Week 3

Frequency of event.Definition of the probability in discrete sample space. Classical definition and applications of probability.

Week 4

Definitions and applcations of Algebra and Borel sigma algebra. Definition of the probability in continuous sample space.

Week 5

Kolmogorov's axioms. Probability space. Properties of the probability measure.

Week 6

Geometrical probability and its applications. Independence of events, multiplication formula and applications of events.

Week 7

Conditional probability and its applcations. Formula of total probability formula. Bayes's formulas and their applications.

Week 8

Mid-term exam

Week 9

Sequance of independent trails. Bernoulli scheme.

Week 10

Limit theorems in Bernoulli scheme (Mouavr-Laplace's local and integral formulas, Poisson's theorem).

Week 11

Law of large numbers. Bernoulli and Poisson theorems.

Week 12

Polynomial scheme.

Week 13

Simple Markov chains.

Week 14

Concept of random variable (assessbility in comparison to algebra) Operations over random variables.

Week 15

Distribution of variables, distribution functions and its basic properties.

Week 16

End-of-term exam

Textbook / Material

1

Akdeniz F. Olasılık ve İstatistik, Ankara Ü., Ankara, 1984,

2

Nasirova T., Khaniyev T. Yapar C., Ünver İ., Küçük Z. Olasılık. KTÜ Matbaası, Trabzon, 2009.

Recommended Reading

1

Kolmogorov A.N. Foundations of the Theory of Probability. New York, 1956.

2

Ceyhan İnal H., Günay S.Olasılık ve matematiksel istatistik, Ankara,1982.

3

Ersoy N., Erbaş S.D. Olasılık ve İstatistiğe giriş, Gazi Ü., Ankara, 1992.