Mathematical Statistics

obligatory courses

The aim of the course is to discuss selected concepts of mathematical statistics and probability calculus. These are the concepts necessary to pass further courses, including econometrics. The first part of the lecture concerns probability calculus. The most important concepts of the probability theory are: random variable, distribution of a random variable, basic characteristics of the distribution of a random variable and types of random variables. In the second part of the lecture selected issues of the theory of statistical inference will be presented, such as point estimation, interval estimation, theory of verification of statistical hypotheses. The aim of this part is to prepare students for independent use of selected statistical procedures.

The student solves the housework, which gives 10% points, and finally passes the written exam, which gives 90% points. The exam consists in solving 5-10 written assignments.

Detailed program:

Part one - elements of the probability theory:

1. Classic probability definition, selected combinatorial schemes. Basic properties of probability.

2. Sample space. Random event. Probability distribution. Independence of events. Definition of a random variable and its types. The probability distribution of a random variable.

3. Cumulative distribution functions of the random variable distribution and its properties. The density function of a continuous type random variable and its properties.

4. The expected value and the variance of the random variable. Other characteristics of the distribution of a random variable. Expected value and variance for the most important distributions. Independence of random variables. Central limit theorem.

Part two - mathematical statistics:

5. Random sample and basic characteristics of its distribution description. Statistical model and the concept of statistics. Sample statistics as examples of statistics. Positional statistic and its distribution, mean and variance from the sample in the normal distribution. Distribution of chi-square, t-student, F-Fisher.

6. Point estimation - the moment method and the method of the greatest credibility. Properties of estimators: estimator load, unloaded estimators. Estimator quality meters, estimator risk with a square loss function, information inequality, estimator efficiency.

7. Interval estimation: the concept of the confidence interval at a given confidence level, confidence intervals for parameters in the normal distribution, asymptotic confidence intervals, confidence interval for the structure index.

8. Verification of statistical hypotheses. The concept of hypothesis, statistical test, significance level and critical area of the test. p-value and test power.

9. Testing hypotheses regarding parameters in a normal distribution. Testing hypotheses regarding the p-structure index. Comparing two and more populations: tests for two expected values, two variances in normal models, one-way analysis of variance. Test for the hypothesis regarding the structure index.

10. Non-parametric tests, chi-square test, chi-square independence test. Summary.

W. Niemiro, Rachunek Prawdopodobieństwa i Statystyka Matematyczna, wyd. SNS, 1999 (część II: Statystyka Matematyczna). [Sygn. Bibl. WNE UW: 33103]

W. Feller, Wstęp do rachunku prawdopodobieństwa, t. I, wyd. II, PWN, Warszawa 1966. [Sygn. Bibl. WNE UW: 11841]

J. Jakubowski, R. Sztencel, Elementarny rachunek prawdopodobieństwa, Warszawa 2001.

L. Gajek i M. Kałuszka, Wnioskowanie Statystyczne, modele i metody, WNT 2000, 1996. [Sygn. Bibl. WNE UW: 31973, 31974 (2000 r.)]

Tablice statystyczne - R. Zieliński, W. Zieliński

- W. Krysicki i in., Rachunek prawdopodobieństwa i statystyka matematyczna w zadaniach, PWN, 1998 (część II: Statystyka Matematyczna). [Sygn. Bibl. WNE UW: 27978/2 (1994 r.), S-9275 a-z (1998 r.), S-8969 a-n, 30479/2]

- H. Kassyk-Rokicka, Statystyka, zbiór zadań, 2005 lub inne wydania

- J. Greń, Statystyka Matematyczna, modele i zadania, PWN, 1978. [Sygn. Bibl. WNE UW: S-1060 b (1976 r.), 15489 (1978 r.)]

A. Boratyńska, Zadania ze statystyki matematycznej, skrypt w ksero na WNE UW

KNOWLEDGE

The student knows and understands selected concepts of probability calculus and mathematical statistics, the most important of which is a random variable, distribution of a random variable, basic characteristics of the distribution of a random variable and types of random variables. Knows the theory of statistical inference, point estimation, interval estimation, the theory of verification of statistical hypotheses. The student knows parametric and nonparametric models for verification of hypotheses regarding theoretical distribution.

SKILLS

The student is able to use the tools of mathematical statistics. He can use selected statistical procedures. Student is able to describe models in formal statistical language. The student is able to use analytical methods to correctly formulate and solve tasks in the field of mathematical statistics. The student is able to construct an unbiased and effective parameter estimator using the chosen method. The student is able to estimate the parameter using the confidence interval. He can verify the hypothesis regarding theoretical distribution.

COMMON SKILLS

The student knows the applications of theories and methods of mathematical statistics in economics and related sciences.

KW01, KW02, KW03, KU01, KU02, KW03, KK01, KK02, KK03

The final result is based on two components - housework and written final exam. The result of the exam gives a formal assessment of the subject (90%). Homework is 10% of the final result. The written exam is about solving 5-10 tasks