Probability and Statistics, First Course

Aim

The aim of the course is to give an introduction to probability and statistics, i.e. to introduce theoretical probability models and to give methods for statistical inference based on observed data. By the end of the course the student should be able to:

• M1 (probability calculations): describe and use models for phenomena affected by chance and carry out probability calculations.
• M2 (stochastic variables): use random variables and their properties to describe and explain random variation.
• M3 (point estimates): use an appropriate random model to describe and analyze observed data, draw conclusions about parameters of interest, and derive point estimates of parameters and analyze their properties.
• M4 (Confidence intervals and hypothesis testing): summarize the principles of drawing conclusions via confidence intervals and hypothesis testing, and construct confidence intervals and conduct hypothesis testing for observed data, report the conclusions and assess certainty.
• M5 (Software): use Matlab or R software to analyze and solve probability and statistics problems.

Content

Probability theory: Sample space, events and probabilities. Combinatorics. Conditional probabilities and independent events. Discrete and continuous random variables, their probability distributions, expectations and variances. Normal, exponential, binomial, Poisson distributions etc. The Central Limit Theorem. Statistics: Point estimation. Properties of estimators. The method of maximum likelihood, the method of moments and the least squares estimation. Confidence intervals. Testing statistical hypotheses. Linear and logistic regression