
Probability Theory and Statistics

Spring/Summer 20192020

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: 8:008:15, the interval 8:158:20 is reserved to gather the results,

test 2: 8:258:40, the interval 8:408:45 is reserved to gather the results,

test 3: 8:509:05, the interval 9:059:10 is reserved to gather the results,

test 4: 9:15 si 9:30, the interval 9:309:35 is reserved to gather the results,

test 5: 9:40 si 9:55, the interval 9:5510:00 is reserved to gather the results,

test 6: 10:05 si 10:20, the interval 10:2010:25 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 online 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.
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 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.
Final grades:
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 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 17, 2020: Introduction, Random experience and random events. Probability function.
 Lecture 2 on February 24, 2020: Conditional probability. Independence. Probabilistic formulas.
 Lecture 3 on March 2, 2020: Multiplication formula. Probabilistic schemata. Discrete random variables.
 Lecture 4 on March 9, 2020: Discrete random variable characteristics. Remarkable discrete distributions. Joint probability distributions.
 Lecture 5 on March 16, 2020: Covariance of random variables. Independent random variables. Inequalities.
 Lecture 6 on March 23, 2020: Random processes. Markov chains. Random walks.
Statistics Lectures:
 Lecture 7 on March 30, 2020: Descriptive statistics. Central tendency. Variability.
 Lecture 8 on April 6, 2020: Continuous Random Variables. The Fundamental Laws. Computer Simulation.
 Lecture 9 on April 13, 2020: Computer Simulation: Monte Carlo Methods.
 Lecture slides in romanian.
 Lecture slides in english.
 video 1 video 2 video 3 video 4 in romanian.
 Laboratory 3 in romanian.
 Laboratory 3 in english.
 Lab scripts
 Tema partea B (ro) Homework part B (en)
In this homework probabilities could be subject to change: e.g. the virus spreading probability 0.15 from B5 should be viewed as a parameter that you can change to 0.25 when calling the function or 0.65, the virus cleaning probability, can be changed to 0.85 or to any other convenient value. (Courtesy of mr. Buliga N., group B1.)
For problem B6 the coordinates (50, 100) must be changed to (100, 50). (Courtesy of mr. Cheptanariu D., group B1.)
 Lecture 10 on April 27, 2020: Randomized Algorithms. Probabilistic Method.
 Lecture 11 on May 4, 2020: Confidence Intervals. Tests of Significance. Proportions Test.
 Lecture 12 on May 11, 2020: 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 18, 2020: Linear Correlation. Linear Regression.