UAIC
Computer Science Department

Probability Theory and Statistics

Olariu Emanuel Florentin

Spring/Summer 2018-2019


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 e-mail 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 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

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 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.

 


Lectures: