3035.18 - Probability theory
Upper secondary school with level A in mathematics
To give the participants an intuitive and rigorous understanding of basic probability theory.
Probability axioms, the inclusion-exclusion principle, conditional probability, independence, Bayes’ theorem, probability distributions, binomial distribution, normal approximation, random sampling, hypergeometric distribution, discrete random variables, expectation, deviation, variance, Markov’s inequality, Chebychev’s inequality, the central limit theorem, geometric distribution, negative binomial distribution, Poisson distribution, continuous random variables, normal distribution, exponential distribution, gamma distribution, survival function, hazard rate, shifting and scaling of one-dimensional continuous random variables, order statistics, uniform distribution, 2-dimensional continuous random variables, Rayleigh distribution, chi-square distribution, conditional distributions, conditional expectation, conditional variance, covariance, correlation, the bivariate normal distribution.
Learning and teaching approaches
Lectures, exercise sessions and independent/individual work.
By the end of the course the participants are expected to be able to: • Apply simple approximation formulas for calculation of probabilities. • Use operations of random variables to obtain new probability distributions. • Analyse combinatorial problems using counting arguments. • Formulate simple probability models from a verbal description. • Use basic concepts from probability theory to solve problems. • Apply the concept of conditional probability to solve problems. • Apply the concepts of conditional distribution and conditional moments to solve problems. • Choose the correct probabilistic model for a real world phenomenon based on its characteristics. • Perform simple calculations with expectation, variance and correlation. • Apply and utilize the relationship between the different ways of expressing a probability distribution. • Calculate distributions and quantities derived from the bivariate normal distribution. • Apply characteristics of discrete and continuous distributions respectively. • Identify and describe probability distributions, including Poisson, binomial, exponential, hypergeometric, gamma, uniform, Rayleigh, chi-square, and the normal distributions.
4-hours written multiple choice test at the end of the course. To be allowed to take the final exam, mandatory tests must be passed during the course. Permitted aids for the final exam: All written material and calculators.
Online teaching material and Jim Pitman: Probability, ISBN: 9780387979748
Toke Meier Carlsen