Probability Theory and Statistics
This course aims to introduce basic topics in Probability Theory and Descriptive and Inferential Statistics.
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 and activity 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.
For arrears to homework Statistics, English groups: 20 of June, in C212, 10:00-11:00
Seminars/laboratories final situations
- 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.
- Lecture 1 on February 18, 2019: Introduction, Random experience and random events. Probability function.
- Lecture 2 on February 25, 2019: Conditional probability. Independence. Probabilistic formulas.
- Lecture 3 on March 4, 2019: Multiplication formula. Probabilistic schemata. Discrete random variables.
- Lecture 4 on March 11, 2019: Discrete random variable characteristics. Remarkable discrete distributions. Joint probability distributions.
- Lecture 5 on March 18, 2019: Covariance of random variables. Independent random variables. Inequalities.
- Lecture 6 on March 25, 2019: Random processes. Markov chains. Random walks.
- Lecture 7 on April 1, 2019: Descriptive statistics. Central tendency. Variability.
- Lecture 8 on April 15, 2019: Continuous Random Variables. The Fundamental Laws. Computer Simulation.
- Lecture 9 on April 22, 2019: Computer Simulation: Monte Carlo Methods.
- Lecture 10 on May 6, 2019: Randomized Algorithms. Probabilistic Method.
- Lecture 11 on May 13, 2019: Confidence Intervals. Tests of Significance. Proportions Test.
- Lecture 12 on May 20, 2019: Tests of Significance. Inferences for the mean of a population: Z-test and T-test. Inferences for two populations: Z-test. Inferences for two variances: F-test.
- Lecture 13 on May 27, 2019: Linear Correlation. Linear Regression.