Probability and Statistics

Aim

The overall goal of the course is to provide an introduction to the mathematical theory of statistical analysis of uncertainty using probability models, with a special focus on problems within computer science and software engineering. The course provides knowledge on how probability models and statistical theory are used to draw conclusions about uncertain quantities based on observed data, make predictions, and make optimal decisions under uncertainty. The statistical inference theory is presented using both a frequentist and a Bayesian approach.

Upon completion of the course, the student should be able to:

• formulate relevant random models for statistical analysis of uncertainty.
• perform probability calculations for stochastic variables and their distributions.
• derive point estimates using the maximum likelihood method and calculate confidence intervals.
• derive a Bayesian posterior distribution for a number of commonly used probability models.
• make predictions and make optimal decisions using probability models.
• implement statistical calculations and simulation with probability models in a modern programming language.

Content

Events and their probabilities. Axioms for probability theory. Interpretations related to concepts in probability theory. Combinatorics. Stochastic variables. Discrete and continuous univariate distributions, such as Bernoulli, uniform, Poisson, normal, t-, gamma, chi-squared, and beta distributions. Multivariate probability distributions, such as multivariate normal, multinomial, and Dirichlet distributions. Marginal and conditional distributions. Expected value and variance. Covariance and correlation. Statistical dependence. Chebyshev's inequality. Law of large numbers. Central limit theorem. Simulation. Introduction to stochastic processes. Markov processes. Descriptive statistics and graphical representation. Point estimation. Sampling distribution. Likelihood function and maximum likelihood method. Bayesian inference. Confidence intervals and Bayesian probability intervals for population means and proportions. Introduction to hypothesis testing. Linear regression. Prediction. Cross-validation. Decision theory. Problem solving in computer science and software engineering.