Schedule

of

„Probability Theory & Mathematical Statistics”

Part one: Probability Theory

1.) Elements of Probability: sample space and events, venn diagrams and the algebra of events, Kolmogorov type of probability space, sample spaces having equally likely outcomes, conditional probability, Bayes’ formula, independent events.

2.) Random variables and its characteristics: definition, types of random variables, probability distribution function, probability mass function for discrete random variables, probability density function for continuous random variables; joint probability distribution function, joint probability mass function, joint probability density function, conditional distribution and independence

3.) Expectations and moments: mean, median, and mode, central moments, variance, and standard deviation, conditional expectation, Chebyshev inequality, moments of two or more random variables, covariance and correlation coefficient, Schwarz inequality.

4.) Some important discrete distributions: Bernoulli trials, binomial distribution, geometric distribution, negative binomial distribution, multinomial distribution, Poisson distribution, special distributions, approximations of the binomial distribution.

5.) Some important continuous distributions: uniform distribution, bivariate uniform distribution, Gaussian or normal distribution, exponential distribution, chi-squared distribution, conditional expectation, the laws of numbers, the central limit theorem.

Part two: Mathematical Statistics

1.) Statistical inference, histogram and frequency diagrams, parameter estimation.

2.) Parameter estimation: samples and statistics, sample mean, sample variance, sample moments, order statistics, quality criteria for estimates, unbiasedness, minimum variance, consistency, sufficiency, methods of estimation, point estimation.

3.) Methods of Estimation: point estimation, interval estimation.

4.) Hypothesis testing (based on rejection region and the P-value): tests concerning the mean of a normal population, case of known variance (the z-test), case of unknown variance (the t-test), testing the equality of means of two normal populations, case of known variances (the paired z-test), case of unknown but equal variances (the paired t-test), case of unknown and unequal variances (the Welch- test), Kolmogorov–Smirnov test.

5.) Linear models and linear regression: Simple Linear Regression; Least Squares Method of Estimation; Properties of Least-Square Estimators; Confidence Intervals for Regression Coefficients.

Requirements: The course ends with a signature and an exam mark.

• Signature condition: Every registered student will have a signature.

• Colloquium condition: The colloquium is written. The exams consist of 6 practical questions. The exam mark is the minimum of 5 and the number of well-answered exercises. The exam is 110 minutes long.

References

1. Sheldon M. Ross: Introduction to Probability and Statistics for Engineers and Scientists, Elsevier Academic Press, 2004.

2. T.T. Soong: Fundamentals of Probability and Statistics for Engineers, John Wiley & Sons Ltd, 2004.

3. Douglas C. Montgomery and George C. Runger: Applied statistics and probability for engineers, John Wiley & Sons, Inc., 2003.

4. Gopal K. Kanji: 100 Statistical Tests, Sage Publications Ltd, 2006

dr. habil. Nutefe Kwami Agbeko

(lecturer)