
Probability Theory and Statistics

Spring/Summer 2024

Summary
This course aims to introduce basic topics in Discrete Probability Theory and Descriptive and Inferential Statistics.
Prerequisites: Knowledge of basic analysis and algebra.
Administrative
Lecturers: Olariu E. Florentin  C212, C building,
phone: 0232 20 15 46, fe dot olariu at gmail dot com
Zalinescu Adrian  C307,
C building
Office Hours: weekly, better by email appointment
Course description: ro  en
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.
 Laboratories. The score comes partially from the exercises solved in the 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.
 For other details see the first lecture (ro)  (en) .
 There is no arrears session exam.
Seminar/laboratory situations:
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.
 Proportion 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.
Probability Theory Lectures:
 Lecture 1 on February 26, 2024: Introduction, Random experience and random events. Probability function.
 Lecture 2 on March 4, 2024: Conditional probability. Independence. Probabilistic formulas.
 Lecture 3 on March 11, 2024: Multiplication formula. Probabilistic schemata. Discrete random variables.
 Lecture 4 on March 18, 2024: Discrete random variable characteristics. Remarkable discrete distributions. Joint probability distributions.
 Lecture 5 on March 25, 2024: Covariance of random variables. Independent random variables. Inequalities.
 Lecture 6 on April 1, 2024: Random processes. Markov chains. Random walks.
Statistics Lectures:
 Lecture 7 on April 8, 2024: Descriptive statistics. Central tendency. Variability.
 Lecture 8 on April 22, 2024: Continuous Random Variables. The Fundamental Laws. Computer Simulation.
 Lecture 9 on April 29, 2024: Computer Simulation: Monte Carlo Methods.
 Lecture 10 on May 13, 2024: Randomized Algorithms. Probabilistic Method.
 Lecture 11 on May 20, 2024: Confidence Intervals. Tests of Significance. Proportions Test.
 Lecture 12 on May 27, 2024: 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 25, 2024: Linear Correlation. Linear Regression.