Mathematical Statistics, First Course

Aim

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

• describe and use models for phenomena influenced by random factors and calculate probabilities;
• use random variables and their properties to describe and explain random variation, use a probability mass function or a density function to calculate probabilities, expected values, variances etc;
• apply an appropriate probabilty model to describe and analyse observed data and draw conclusions concerning interesting parameters by using point estimation, confidence intervals and hypothesis testing;
• analyse the relationships between two or several variables by using simple or multiple linear regression models and discuss the adequacy of the models;
• use probability models and statistical methods in applications from science and engineering and evaluate the results;
• use software for certain types of statistical analyses.

Content

Probability: Sample spaces, events and probabily. Conditional probability. Independent events. Random variables, expected value, variance and standard deviation; especially normal, exponential, binomial and Poisson distribution. The central limit theorem. Statistics: Methods for parameter estimation. Confidence intervals in connection with one or several samples from the normal distribution, for proportions, in connection with the Poisson distribution etc. Tests of hypothesis in particular chisquare tests. Simple and multiple linear regression. Histograms.