Probability Theory and Mathematical Statistics
- Studying the theoretical foundations of probability theory and mathematical statistics, their practical development through the construction of mathematical models and solving statistical problems
- Understanding the types of practical problems, including those arising in sociology, which can be solved using statistical methods, and the ability to use the knowledge gained to solve them
- Ability to work with programs for mathematical calculations
- Deepening and expanding the range of knowledge about applied mathematical methods
- Mastering modern methods of data analysis, for example, basic skills of data research using statistical packages such as R (S-plus)
- Is able to describe a random experiment and the set of all outcomes.
- Is able to apply Classical Formula of probability.
- Knows how to apply Bernoulli formula.
- Knows the law of total probability.
- Is able to calculate posterior probabilities by Bayes’ formula.
- Is able to construct random variables describing a given random experiment.
- Is able to calculate the expected value and variance of these random variables.
- Knows the main families of discrete and continuous random variables.
- Calculates any probability for Normal Distribution.
- Is able to approximate probabilities of large number of similar events by CLT.
- Knows what is a population and what is a sample from population.
- Calculates sample mean, sample variance, unbiased sample variance, sample proportion and quantiles.
- Is able to calculate sample mean, sample variance, unbiased sample variance, sample proportion and quantiles in R.
- Knows how to construct confidence intervals for the means and proportions.
- Is able to show connection with the CLT.
- Knows when and why one should use Student instead of Normal distribution for CI.
- Knows the main approaches of hypotheses testing.
- Is able to construct the null and the alternative hypothesis.
- Is able to make a statistical inference by the significance level or by p-value.
- Introduction to Probability. Independence. Bernoulli trials.Introduction to Probability • History of Probability • Probability and Data Analysis • Random experiment. Outcomes. Events. • Operations with Events • Statistical definition of probability Properties of Probability • Axiomatic definition of probability • Classical formula of probability for equally likely outcomes. • Inclusion-exclusion formula • Hypergeometric distribution formula • Birthday paradox • Independent and dependent events Bernoulli trials • Independent experiments • Formula for the number of successes. Proof. • Banach's matchbox problem • Probability of the first success in the k-th trial
- Conditional probability. Bayes formula.Conditional probability • Definition. Illustration via contingency tables • Multiplication formula for probabilities • Independence via conditional probability • Generalization of Multiplication formula for k events. Chain rule. Law of Total Probability • Law of Total Probability for two events. Proof. • Group of jointly exhaustive events • Generalization of the law of total probability • Monty Holl Paradox Bayes’ Theorem • Bayes’ formula • Prior and posterior probabilities • Example from bookmaking company • Odds ratio • Conditional independence
- Discrete and continuous random variables. Theirs numerical characteristics.Discrete random variables • Definition. Main properties • Probability mass function • Binomial random variable. Binomial formula in math analysis. • Geometric variable • Hypergeometric random variable • Poisson random variable. Rare events. Poisson limit theorem Numerical characteristics of random variables • Expectation. Variance. Standard deviation. Theirs properties. • K-th moment • Mode. Median. • St. Petersburg paradox Continuous Random Variables • Cumulative distribution function • Density of continuous random variable • Uniform, Gaussian (Normal), Exponential random variables • Formulas for expectation and variance • Independence and dependence of random variables. Joint distribution.
- Normal distribution and Limit laws: CLT, LLN.Normal Distribution • Family of normal distributions • Standardization. Z-score • 65-95-99.7-rule (3-sigma rule) Limit theorems • CLT. Examples • LLN. Examples More limit theorems • CLT for proportions • De Moivre-Laplace theorem
- Introduction to statistical analysis. Sample. Statistical population. Point estimation.Statistics. Preliminaries • Main definitions: population, sample, representative sample • Frequency histogram • Empirical distribution function • Probability vs Statistics Properties of Point estimates • Unbiasedness • Consistency • Quantiles. • Sample mean, sample variance, unbiased sample variance, sample proportion • Outliers • Correlation • Introductory seminar for R
- Interval estimationConfidence intervals • Point estimates vs Interval estimates • Confidence interval for a mean (variance is known) • Student T-distribution • Confidence interval for a mean (variance is not known) • One-sided confidence intervals • Confidence intervals for proportions
- Hypothesis testingStatistical Hypothesis Testing • Statistical hypothesis • Statistical test • Type 1 and Type 2 errors • Significance level. Rejection Region. • P-value Hypotheses for a mean of normally distributed data • One-sample Z-test. Connection with the CLT. • Two-sample T-test • Applications: A/B testing, Model validation problems, Double blind experiment • Connection with the confidence intervals Homogeneity hypothesis • Mann-Whitney rank test • Two-sample Kolmogorov test • Applications: A/B testing, Model validation problems, Double blind experiment
- Interim assessment (4 module)0.113 * Activity + 0.186 * Average of the Class assignments + 0.25 * Exam + 0.075 * Self-study report + 0.188 * Test 1 + 0.188 * Test 2
- Deep, R. (2006). Probability and Statistics : With Integrated Software Routines. Amsterdam: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=196153
- Young, G. A., & Smith, R. L. (2005). Essentials of Statistical Inference. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=138968
- Bruce, P. C. (2014). Introductory Statistics and Analytics : A Resampling Perspective. Hoboken, New Jersey: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=923330