Probability, First Course

Aim

The aim of the course is to provide an introduction to the mathematical modelling of random experiments. The emphasis is on methods applicable to problems in engineering, economy and natural sciences. After completing the course the student should have the knowledge and skills required to:

• identify experimental situations where random influence may affect the results.
• construct relevant probabilistic models for simple random experiments.
• describe basic concepts and theorems of probability theory, e.g., random variable, distribution function and the law of total probability.
• compute important quantities in probabilistic models, e.g., probabilities and expectations.

Content

• Definition of probability
• Combinatorial methods
• Conditional probability and Bayes rule
• Discrete random variables, probability function, cumulative distribution function.
• Expected value, variance, covariance, correlation
• Special examples: Bernoulli, Binomial, Geometric, Hypergeometric, Negative Binomial, Poisson, and applications.
• Joint probability functions, conditional probability function, conditional expectation.
• Continuous random variables
• Special distributions: the exponential distribution, the normal distribution.
• Sampling and the Central Limit Theorem.
• The Poisson Process and applications.
• Scenarios illustrating applications of probability and statistics.