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Regular version of the site

Probability Theory and Mathematical Statistics

2019/2020
Academic Year
ENG
Instruction in English
5
ECTS credits
Course type:
Compulsory course
When:
2 year, 1, 2 module

Instructor

Course Syllabus

Abstract

Probability Theory and Mathematical Statistics is a core mathematical subject taught to the second year students in the 1st and 2nd academic modules. The material is split between probability theory and statistics almost evenly. The course covers classical probability topics from basic probability to limit theorems. More attention is paid to the conditional moments of multivariate random variables. Depending on the available time more advanced topics such as random walks, the Poisson process and Markov chains may be considered. The statistical section starts with the descriptive techniques but quickly switches to the inferential methods as they are more mathematically involved and require more eorts to explain. The topics covered here include sampling distributions, point and interval estimates, hypothesis testing. We conclude with a univariate and, if time permits, multivariate regression. Throughout the course a certain balance between mathematical rigor and clarity is maintained. Sometimes this dilemma is resolved in favor of illustrative examples which help students capture the main ideas and use them in practice rather than focus on blind memorizing the derivations. However, we find it instructive to provide the tractable proofs whenever it makes pedagogical or some other sense. The course is taught in English and worth 5 credits.
Learning Objectives

Learning Objectives

  • Probability theory and Statistics provides an essential basis in probability theory and mathematical statistics and educates students how to use these principles in practice. Another implicit purpose of PT&Stat is to guide the rst steps in students' research by suggesting more challenging topics and problems to the interested students. This kind of activity develops self-study skills and critical thinking, emphasizes importance of working with literature and provides many more helpful skills
Expected Learning Outcomes

Expected Learning Outcomes

  • Be aware of different definitions of probability, the axioms of probability and their use for derivation of major probabilistic relationships. Know the basic counting methods and principles of combinatorics.
  • Be able to use the concept of conditional probability, law of total probability, notion of independence, collectively exhaustive events.
  • Understand what is meant by a random variable. Know how to use the probability mass functions for calculating the basic characteristics of a discrete random variable (expected value, variance). Be aware of commonly used discrete distributions.
  • Know and be able to use alternative ways of describing a continuous random variable - probability density function and cumulative distribution function. Know how to calculate basic characteristics of a continuous random variable. Be aware of commonly used continuous distributions.
  • Know how to work with a multivariate random variable using the joint probability distribution. Be able to detect the indepedent random variables, calculate the marginal and conditional distributions, covariance and correlation between the variables.
  • Understand the role of limit theorems in the probability theory. Be able to use the Markov and Chebyshev inequalities in practice. Be able to implement the central limit theorem.
  • Be able to use the methods of descriptive statistics to summarize and visualize the raw data.
  • Be able to derive and use the main sampling distributions - for mean, proportion and variance
  • Be able to estimate the main characteristics of a random variable by means of point or interval estimates. Know how to construct and interpret the confidence intervals for mean, proportion and variance.
  • Understand the principles of hypothesis testing. Be able to perform tests for population mean, proportion and variance.
  • Be able to build, assess the quality and make predictions with the linear regression model.
Course Contents

Course Contents

  • Axioms of probability
    The definition of probability. Interpretations of probability. Counting methods. Combinatorial analysis: permutations, combinations, Partitions, multinomial coefficients. Sampling with and without replacement. Sample space and events. Axioms of probability. Probability as a continuous set function. Venn diagrams.
  • Conditional probability and independence
    Conditional probability. Independent events, mutually exclusive events. Exhaustive events. The law of total probability. Bayes' theorem. Probability trees.
  • Discrete random variables
    Random variables. Discrete and continuous distributions. Cumulative distribution function and its properties. Probability mass function. Common discrete distributions: uniform, binomial, Poisson distributions. Sequences of Bernoulli trials. Expected value, variance and their properties. Functions of a random variable
  • Continuous random variables
    Properties of continuous random variables. Prob- ability density function. Expectation and variance of continuous random variables. Common continuous distributions (uniform, exponential, normal, chi-squared, F-distribution, Student's t- distribution) . The normal approximation of the binomial distribution.
  • Jointly distributed random variables
    Joint distributions of random variables. Independent random variables. Expected values, covariance. Sums of independent random variables. Conditional distributions. Marginal distributions. Conditional expectation and variance. Joint probability distribution of functions of random variables. Properties of multivariate normal distribution
  • Limit theorems
    Markov's and Chebyshev's inequalities. Laws of large numbers for Bernoulli summands. The Poisson limit theorem. Weak and strong laws of large numbers. The central limit theorem. Convergence in probability.
  • Methods of descriptive statistics
    The nature of statistics. Population, bias, parameters, sampling. Quantitative and categorical variables. Data presentation: dot plot , histogram, steam-and-leaf. Measures of location: mean, median, mode. Quartiles and percentiles. Measures of spread: range, variance and standard deviation, interquartile range
  • Ideas of sampling and sampling distributions
    Sampling distributions. Sampling distributions for the mean. Distribution of the sample variance
  • Point and interval estimates
    Properties of estimators: bias, variance, mean squared error. Consistency. Finding estimators: method of moments, least squares, maximum likelihood. Confidence intervals for the mean of a normal population. Intervals for mean dierences (paired, unpaired samples) . Confidence intervals for proportions (single proportion, differences between proportions) . Confidence intervals for variance.
  • Hypothesis testing
    Null and alternative hypotheses. One- and two-sided alternative hypotheses. Test statistics and critical regions, signicance levels. Type I and Type II errors. p-values. Level and power of the statistical test. Testing hypotheses about population means and proportions. Link to condence intervals. Two-sample tests. Testing hypotheses about population variances. Categorical data and nonparametric methods: contingency tables and the chi-squared test . Goodness-of-t tests
  • Linear regression model
    The simple linear regression model. Least squares method. Coecient of determination. Means and variances for slope and intercept. Interval estimates for tted values. Prediction with the regression model. Multiple Regression. The model for multiple linear regression. Least squares tting. Collinearity. Diagnostic plots.
Assessment Elements

Assessment Elements

  • non-blocking In-class tests
  • non-blocking Home assignments
  • non-blocking Final Exam
  • non-blocking in-class activity
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.4 * Final Exam + 0.18 * Home assignments + 0.03 * in-class activity + 0.39 * In-class tests
Bibliography

Bibliography

Recommended Core Bibliography

  • Biswas, D. (2019). Probability and Statistics: Volume I. [N.p.]: New Central Book Agency. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2239779
  • Blitzstein, J. K., & Hwang, J. (2019). Introduction to Probability, Second Edition (Vol. Second edition). Boca Raton, FL: Chapman and Hall/CRC. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=nlebk&AN=2024519

Recommended Additional Bibliography

  • Balakrishnan, N., Koutras, M. V., & Konstantinos, P. (2019). Introduction to Probability : Models and Applications. Hoboken, NJ: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2097342
  • Ghahramani, S. (2018). Fundamentals of Probability : With Stochastic Processes (Vol. Fourth edition). Boca Raton, FL: Chapman and Hall/CRC. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1875108
  • Linde, W. (2017). Probability Theory : A First Course in Probability Theory and Statistics. [N.p.]: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1438416
  • Linton, O. B. (2017). Probability, Statistics and Econometrics. London, United Kingdom: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1200673
  • Samuel Goldberg. (2013). Probability : An Introduction. [N.p.]: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152975