 ## Olariu Emanuel Florentin

### Spring/Summer 2019-2020

The score T2 (from statistics laboratories) will be composed from 20 points (the presence) + 40 points (the homeworks)

Arrears and enhancements: students can choose the Probability theory tests to take and the Statistics homeworks to present in order to increase their grades.

Probability theory tests on 18 of June:

• halfyear B and E3, E4 groups on Google Classroom coordinated by prof. A. Zalinescu;

• halfyear A and E1, E2 groups on Zoom coordinated by prof. E. F. Olariu.

The schedule for Probability theory tests is:

• test 1: 8:00-8:15, the interval 8:15-8:20 is reserved to gather the results,

• test 2: 8:25-8:40, the interval 8:40-8:45 is reserved to gather the results,

• test 3: 8:50-9:05, the interval 9:05-9:10 is reserved to gather the results,

• test 4: 9:15 si 9:30, the interval 9:30-9:35 is reserved to gather the results,

• test 5: 9:40 si 9:55, the interval 9:55-10:00 is reserved to gather the results,

• test 6: 10:05 si 10:20, the interval 10:20-10:25 is reserved to gather the results.

Statistics homeworks will be sent between 16 and 17 of June like follows:

• halfyear B and E3, E4 groups to prof. A. Zalinescu at one of these two addresses: adrian.zalinescu@info.uaic.ro or adrian.zalinescu@gmail.com;

• halfyear A and E1, E2 groups on to prof. E. F. Olariu at one of these two addresses: olariu@info.uaic.ro sau fe.olariu@gmail.com

Students from group X may choose an on-line meeting for taking the tests and an address from above to sent their homeworks.

## Summary

This course aims to introduce basic topics in Probability Theory and Descriptive and Inferential Statistics.

#### Lectures: weekly in C403 (in english) and C112/C2 (in romanian)

Lecturer:  Olariu E. Florentin- C212, C building, phone: 0232 20 15 46, olariu at info dot uaic dot ro

Office Hours: weekly, better by e-mail appointment

• Seminars. The score comes from six small tests one on each seminar - this score must be at least 30 (from a maximum of 6x10=60) points. Those who fail to receive at least 30 points cannot pass the course and must retake all the tests in the arrears session.

• Laboratories. The score comes partially from the presence in class (20 points) and partially from homeworks (40 points) whose end-terms are in the last week of the semester. This score must be at least 30 (from a maximum of 20+40=60) points. Those who fail to receive at least 30 points cannot pass the course and must retake all the homeworks in the arrears session.
• For other details see the first lecture.

Prerequisites: Knowledge of basic analysis and algebra.

• After the enhancements and arrears exam: halfyear A, E1, E2 and X group
• Halfyears A, B, E1, E2, E3, E4 and X group
• Seminars/laboratories final situations:

• After the enhancements and arrears exam: halfyear A, E1, E2 and X group
• Halfyears A, B, E1, E2, E3, E4 and X group

• Bibliography:

• Bertsekas, D. P., J. N. Tsitsiklis, Introduction to Probability, Athena Scientific, Belmont, Massachusetts, 2002.
• Gordon, H., Discrete Probability, Springer Verlag, 2010.
• Lipschutz, S., Theory and Problems of Probability, Schaum's Outline Series, McGraw Hill, 1965.
• Ross, S. M., A First Course in Probability, Prentice Hall, 5th edition, 1998.
• Stone, C. J., A Course in Probability and Statistics, Duxbury Press, 1996.

• Freedman, D., R. Pisani, R. Purves, Statistics, W. W. Norton & Company, 4th edition, 2007.
• Johnson, R., P. Kuby, Elementary Statistics, Brooks/Cole, Cengage Learning, 11th edition, 2012.
• Shao, J., Mathematical Statistics, Springer Verlag, 1998.
• Spiegel, M. R., L. J. Stephens, Theory and Problems of Statistics, McGraw Hill, 3rd edition, 1999.

List of Topics (weekly updated):

• Introduction. Random experience.
• Random (elementary) events, probability function.
• Conditional probability, independent random events, conditional independence.
• Total probability formula, Bayes formula, conditional version of the total probability formula.
• Multiplication formula. Probabilistic schemata: hypergeometric, Poisson, binomial, geometric.
• Distribution of a discrete random variable.
• Expectation and variance of a discrete random variable.
• Remarkable discrete distributions: uniform, Bernoulli, binomial, geometric, Poisson, hypergeometric.
• Joint probability distribution.
• Covariance and independence of random variables.
• Markov and Chebyshev inequalities.
• Chernoff bounds. Hoeffding bounds.
• Discrete Markov chains. Steady states, long term behavior. Random walks.

• Vocabulary of statistics. Descriptive statistics, variable, graphical representations.
• Central tendency: mean, median, mode. Quartiles.
• Variability measures: variance, standard deviation, interquartile range. Outliers.
• Continuous random variables. Density and distribution functions. Remarkable continuous distributions.
• Fundamental laws: Law of Large Numbers (LLN) and Central Limit Theorem (CLT).
• Computer simulation. Illustrations of LLN and CLT.
• Computer simulation: Monte Carlo methods.
• Estimating lengths, areas, and volumes. Monte Carlo integration. Estimating probabilities.
• Randomized algorithms. Las Vegas and Monte Carlo algorithms.
• Probabilistic method: satisfiabilty and graph theory applications.
• Inferential statistics. Point and interval estimation - confidence intervals.
• Statistical hypotheses testing. Errors, significance level and the power of the test.
• Proportions test. One- and two-tailed tests.
• Z-test for the mean of a population with known variance.
• T-test for the mean of a population with unknown variance.
• Z-test for the means of two population with known variances.
• F-test for the ratio of variances.
• Linear correlation. The correlation coefficient and the standard deviation line.
• Linear regression. Regression line.

Probability Theory Lectures: