# International College of Economics and Finance Syllabus Probability Theory and Introductory Statistics

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1 International College of Economics and Finance Syllabus Probability Theory and Introductory Statistics Lecturer: Mikhail Zhitlukhin. 1. Course description Probability Theory and Introductory Statistics is a two-semester course for rst-year students of the ICEF specializing in economics. The course is taught in Russian and English. The course is devoted to basic notions of statistics: data collection methods, the notion of a population and a sample, descriptive statistics, statistical methods of parameter estimation and hypothesis testing, regression models, etc. The course also includes topics from Probability Theory, which are essential for a consistent delivery. 2. Course objectives The main objective of the course is to provide students with knowledge of basic statistics. By the end of the course the students should understand the subject of statistics and master its basic methods. They acquire skills of primary data analysis (how to nd a mean, a median, a standard deviation and other descriptive statistics), graphical representation of data (histograms, stem-and-leaf plots, dot plots, box plots). The students learn how to formulate and solve typical problems of basic statistics: point and interval parameter estimation, hypothesis testing, correlation analysis, regression models. An essential part of the course is devoted to basics of Probability Theory, which serves as a foundation of statistics. The students should understand the notion of a probability space, a random event, the probability of an event. They should know how to compute the probability of a complex event, solve basic combinatorial problems, understand the formula of total probability and the Bayes formula. The students should have a clear understanding of the concept of a random variable and its distribution. They should also understand the meaning of the Law of Large Numbers and the Central Limit Theorem. In classes students both solve theoretical problems and perform computer tasks with real data, develop practical skills and intuition. By the end of the course, students should understand the theory of statistical methods and be able to apply them in practice. In the course students obtain the amount of knowledge sucient for the AP Statistics test. 3. Methods The following methods and forms of study are used in the course: lectures (2 hours a week), classes (2 hours a week), weekly home assignments, oce hours, self-study. The course consists of 58 hours of lectures and 58 hours of classes. 4. Grade determination The students sit a written mid-term exam in the rst module and a nal semester exam in the end of the second module. The exams include multiple choice and free response questions. In the fourth module the students sit a mid-term exam of a similar form. After that the students have the AP Statistics exam. 1

3 6. Course outline 1. Elements of probability theory (WW, Ch. 3, 4) 1.1 Experiment with random outcomes. Notion of probability. 1.2 Space of elementary outcomes as a mathematical model of an experiment with random outcomes. Algebra of events. Disjoint events. 1.3 Probability in a space of elementary outcomes. Classical probability. Elementary combinatorics. Probability of the sum of events. 1.4 Conditional probability. Probability of the product of events. Independent events. 1.5 The formula of total probability. The Bayes formula. 2. Discrete random variables (WW, Ch. 3, 4) 2.1 Examples of discrete random variables. Distribution of a discrete random variable. Relative frequencies and cumulative frequencies. 2.2 Mean value (expectation). Variance. Standard deviation. 2.3 Sequence of independent experiments. The binomial distribution. The geometric distribution. 3. Continuous random variables (WW, Ch. 3, 4) 3.1 Examples of continuous random variables. Distribution function. Distribution density. Mean value (expectation). Variance. Standard deviation. 3.2 The normal distribution, its properties. Normal distribution tables. 3.3 Linear transformations of random variables. 4. Two-dimensional distributions (WW, Ch. 5) 4.1 Joint distribution of two random variables. Marginal distribution. Conditional distribution. Conditional expectation. 4.2 Independent random variables. 4.3 Covariation coecient. Correlation as a measure of the linear relationship between two random variables. Uncorrelated and independent random variables. The expectation and the variance of a linear combination of random variables. 5. Limit theorems 5.1 The Law of Large Numbers. 5.2 The Central Limit Theorem. Normal approximation of the binomial distribution. 6. Basic data analysis and descriptive statistics (WW, Ch.2) 6.1 Graphical representation of data. Dot plots. Steam-and-leaf plots. Histograms. 6.2 Characteristics of data. Outliers. Clusters. Histogram shape. 6.3 Descriptive statistics. Measures of center of a distribution: arithmetic average, median, mode. Measures of dispersion of a distribution: range, mean-square deviation, interquartile range, average absolute deviation, average relative deviation. Representation of data with box plots. 6.4 Transformation of basic statistics under a linear transform of data. 6.5 Measures of location in a sample: quartiles, percentiles, z-scale. 6.6 Computations with grouped data. 7. Data collection, planning and conducting an experiment (WW, Ch.1) 7.1 Methods of data collection: census, sample survey, experiment, observational study. 7.2 Population, sample, random sample. 7.3 Sources of bias in sampling and surveys. 7.4 Types of sampling: simple random sampling, stratied random sampling, cluster sampling. 7.5 Planning and conducting an experiment. 7.6 Control groups, random assignments, replication. 7.7 Sources of bias in experiments. Mixing factors, placebo eect, blinding. 7.8 Completely randomized design. Block design. 3

4 8. Sampling distributions (WW, Ch. 6) 8.1 The distribution of the sample mean and the sample proportion. 8.2 The distribution of the dierence of two proportions. The distribution of the dierence of two independent sample means. 8.3 Student's t-distribution, the chi-squared distribution. 9. Point parameter estimation (WW, Ch. 7) 9.1 Point estimation of population parameters. Examples of point estimates: sample mean and sample variance. 9.2 Properties of estimates: unbiasedness, eciency, consistency. 9.3 Estimates of mean and variance. 9.4 Estimates of proportion. 10. Interval parameter estimation (WW, Ch. 8) 10.1 The notion of a condence interval. The condence interval for the mean of a population. Normal approximation for large samples. Small samples (Student's distribution) Condence intervals for the dierence of two population means (independent and matched samples) Condence intervals for the dierence of two proportions Two-sided and one-sided condence intervals. 11. Hypothesis testing (WW, Ch. 9) 11.1 Hypothesis and statistical test. Test for a population mean. Using condence intervals and test-statistics Two-sided and one-sided tests. P -value Type I errors and type II errors. Signicance and power of a test Standard tests: population mean, population proportion, dierence of two independent and matched samples, dierence of proportions Pearson's chi-squared test. Contingency tables. 12. Pair regression (WW, Ch. 11, 12) 12.1 XY plot. Fitting a line. Ordinary least squares Transformations into a linear model Outliers Fitted values Errors and residuals Statistical properties of regression estimates. Condence interval for the slope. Testing hypothesis for the slope. 4

5 7. Distribution of hours for topics and types of work No. Topics Number of hours lectures classes 1 Elements of Probability Theory Discrete random variables Continuous random variables Two-dimensional distributions Limit theorems Basic data analysis Data collection, planning and conducting an experiment Sampling distributions Point parameter estimation Interval parameter estimation Hypothesis testing Pair regression 6 6 TOTAL

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