# Syllabus

### I. Course description:

The course will provide a rigorous foundation in the principles of probability and mathematical statistics underlying statistical inference in the eld of economics and business. This course is a prerequisite for the lecture Advanced Statistics II, which focuses on the methods of statistical inference including parameter estimation and hypothesis testing. Furthermore, the course provides the foundation for the specialization courses in statistics and econometrics (Time Series Analysis, Statistics for Financial Markets, Microeconometrics, Multivariate Statistics, etc.). In the accompanying tutorial, the students will practice the use of the methods presented in the lecture.

### II. Prerequisities:

The course assumes knowledge of the topics taught in the statistics and mathematics
courses of the Bachelor program.

Written exam.

### IV. Outline:

1. Elements of Probability Theory:

1.1  Sample Space and Events

1.2  Probability

1.3  Properties of the Probability Function

1.4  Conditional Probability

1.5  Independence

1.6  Total Probability Rule and Bayes' Rule

2. Random Variables and their Probability Distributions:

2.1  Univariate Random Variables

2.2  Univariate Cumulative Distribution Functions

2.3  Multivariate Random Variables

2.4  Marginal Distributions

2.5  Conditional Distributions

2.6  Independence of Random Variables

3. Moments of Random Variables:

3.1  Expectation of a Random Variable

3.2  Expectation of a Function of Random Variables

3.3  Conditional Expectation

3.4  Moments of a Random Variable

3.5  Moment-Generating Functions

3.6  Joint Moments and Moments of Linear Combinations

3.7  Means and Variances of Linear Combinations of Random Variables

4. Parametric Families of Density Functions:

4.1 Discrete Density Functions

4.2  Continuous Density Functions

4.3  Normal Family of Densities

4.4  Exponential Class of Distributions

5. Basic Asymptotics:

5.1  Convergence of Number and Function Sequences

5.2  Convergence Concepts for Sequences of Random Variables

5.2.1  Convergence in Distribution

5.2.2  Convergence in Probability

5.2.3  Convergence in Mean Square

5.3  Weak Laws of Large Numbers

5.4  Central Limit Theorems

5.5  Asymptotic Distributions of Functions for Asymptotically Normally Distributed Random Variables

6. Sample Moments and their Distributions

6.1  Random Sampling

6.2  Empirical Distribution Function

6.3  Sample Moments

6.4  Sample Mean and Variance from Normal Random Samples

6.5  Probability Density Functions of Functions of Random Variables

### V. Literature:

This course is based on the following two textbooks:

• Mittelhammer, R.C. (1996). Mathematical Statistics for Economics and Business. New-
York: Springer-Verlag.
• Mood, A.M., Graybill, F.A. und D.C. Boes (1974). Introduction to the Theory of Stati-
stics
. Boston: McGraw-Hill. 3. Edition.

### Further useful textbooks are:

• Casella, G. und R. Berger (2002). Statistical Inference. Pacic Grove: Duxbury, 2. Edition.
• Dudewicz, E.J. und S.N. Mishra (1988). Modern Mathematical Statistics. New-York: John Wiley & Sons.
• Fish, M. (1989). Wahrscheinlichkeitsrechunung und Mathematische Statistik. Berlin: VEB Deutscher Verlag der Wissenschaften. 11. Auflage.
• Hogg, R.V. und R. Craig (1995). Introduction to Mathematical Statistics. Prentice Hall: London, 5. Edition.
• Rohatgi, V.K. und A.K. Saleh (2001). An Introduction to Probability Theory and Mathematical Statistics. New-York: John Wiley & Sons, 2. Edition.