
Probability Theory and Statistics

Spring/Summer 20182019

Summary
This course aims to introduce basic topics in Probability Theory and Descriptive and Inferential Statistics.
Administrative
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 email appointment
Grading:
 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 endterms 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, 10:0011:00
Seminars/laboratories final situations
Final grades
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 twotailed tests.
 Ztest for the mean of a population with known variance.
 Ttest for the mean of a population with unknown variance.
 Ztest for the means of two population with known variances.
 Ftest for the ratio of variances.
 Linear correlation. The correlation coefficient and the standard deviation line.
 Linear regression. Regression line.
Lectures:
 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: Ztest and Ttest. Inferences for two populations: Ztest. Inferences for two variances: Ftest.
 Lecture 13 on May 27, 2019: Linear Correlation. Linear Regression.