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Probability Theory and Mathematical Statistics

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This document provides information about a Probability Theory and Mathematical Statistics course taught at KNITU, Russia. It includes details about the course such as the number of students, preliminary courses required, distribution of working time, topics covered in lectures and workshops/laboratories. It also compares the methodology and topics studied in this course to a similar course taught at TUT, Finland. Key differences highlighted include the use of Matlab at TUT and more emphasis on practical work/tutorials versus lectures. Overall competencies covered are also summarized and compared between the two courses based on the SEFI framework.

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PROBABILITY THEORYPROBABILITY THEORY AND MATHEMATICALAND MATHEMATICAL STATISTICSSTATISTICS Svetlana Medvedeva 2 semester - spring semester of the first year of a bachelor's degree Bachelor's program "Computer Science and Engineering"
Information about courseInformation about course Number of groups – 6 (80 students) Number of people in group - 11-16. Of these males - 85%, females – 15% Preliminary courses: Algebra and geometry Mathematical analysis Discrete mathematics 2
Distribution of working timeDistribution of working time 3
Laboratory workshopLaboratory workshop 4 № № themes Name of the laboratory works Labor Input (h.) 1 2.1 Distribution laws of random variables 2 2 3.1 Study of a sample of measurements and build some variation. Exclusion of gross errors of measurement 4 3 3.2 Construction and research of the evaluation function and the probability density of the random variable 4 4 3.3 Building and research of point and interval estimates of mathematical expectation and variance. 4 5 3.4 Statistical hypothesis testing on the distribution of the random variable. Kolmogorov Criterion. Pearson Criterion 4
Topics for workshopsTopics for workshops 5 № п/п № theme s Topic name Labor Input (h.) 1 1.1 Actions with events. Classical and geometric definition of probability. Conditional probability. 2 2 1.1 Formulas of addition and multiplication of probabilities 2 3 1.1 Full probabilities and Bayesian formulas. Bernoulli's Formula 2 4 1.2 The laws of distribution of discrete. Actions with tables of distributions. 2 5 1.2 Function and distribution density of continuous random variable 2 6 1.2 Numerical characteristics of random variables 2 7 2.1 Distribution law and numerical characteristics of a system of two random variables The correlation time of the system of two random variables. The correlation coefficient 2 8 2.1 The conditional distribution of a discrete random variable. Conditional mathematical expectation and variance. Independence of continuous random variables. 2 9 2.2 The law of large numbers. Chebyshev's Inequality 2
Comparison with methodologyComparison with methodology of TUTof TUT 6 (KNITU, RUSSIA) (TUT, Finland) (TUT, Finland) Selective/mandatory Mandatory Mandatory Mandatory method of teaching Blended Blended Course SEFI level (Core 0, 1, 2 or 3) 0,1,2 0,1,2 0,1,2 Amount of credits 5 4 4 Duration 18 week 7 weeks 7 weeks Lectures 36 28 28 Practice 18 14 14 Laboratory work / tutorials 18 0 0 Homework (% mandatory) 35 39 39 Exam preparation 36 27 27 Exam 2 4 5 Total (number of hours / credits) 180 / 5 108 / 4 108/4 Distribution of study time Probability Theory and Mathematical Statistics
Comparison with methodologyComparison with methodology of TUTof TUT 7 Comparison of topics studied Probability Theory (KNITU, RUSSIA) Probability Theory (TUT, Finland) 1. Random events. 1. Concept of Probability and Calculus Basic concepts of probability theory. Events. Experiments.. Elementary events. Random variable, sample space, event, classical probability, addition theorem, multiplication theorem, conditional probability, total probability, Bayes formula, independence of events 2. Random variables. 2. Probability Distributions Discrete random variables. Continuous random variables. Distribution function. Density distribution. Numerical characteristics of random variables. Mathematical expectation, variance, moments, quantile, the significance level. Basic laws of distribution of random variables. Discrete and continuous distributions, density and cumulative distribution functions, variance and standard deviation, binomial, Poisson distribution, normal distribution, t-, F, and khi^2 distributions 3. Systems random variables. 3. Joint Distributions Random vectors and their distribution. Discrete two- dimensional random variable, its distribution law. Continuous two-dimensional random variable, its distribution law. Conditional distributions. Dependent and independent random variables, the correlation Discrete and continuous distributions, marginal distributions, independency, covariance, correlation, principles of statistical testing, Fundamental limit theorem 4. Limit theorems of probability theory.   Random sequences. Types of convergence of random sequences. Characteristics of the law of large numbers. Chebyshev inequality. Chebyshev's theorem, Bernoulli. The central limit theorem.  
Comparison with methodologyComparison with methodology of TUTof TUT 8 Comparison of topics studied Mathematical Statistics (KNITU, RUSSIA) Mathematical Statistics (TUT, Finland) 1. Initial statistical analysis of the results of measurements of the random variable. 1. Fundamental sampling distributions and data descriptions Definition of a random sample. Priori and a posteriori sampling. Order statistics. Blunders measurements and methods of their elimination. Building interval statistical series. Polygon and the histogram. Empirical distribution Random Sampling, Some Important Statistics, Data Displays and Graphical Methods, Sampling distributions, Sampling distributions of means, The sampling distribution of the sample variance, The sampling distribution of the sample, t-Distribution, F- distribution 2. 1. Spot Parameter Estimation of distributions of random variables. 2. One- and two-sample estimation Optimality criteria point estimation parameters: consistency, unbiasedness, efficiency, sufficiency. Point estimates of the expectation and variance and their properties. "Point Estimation and Interval Estimation, Single Sample: Estimating the Mean," Prediction intervals, tolerance limits, Two Samples: Estimating the Difference between two means, Paired observations, estimating a proportion, "Single sample: Estimating the variance, Two samples: Estimating the ratio of two variances" 2.2. Interval estimation of the distribution parameters. Concept of confidence interval and confidence probability. Interval estimates the expectation and variance.
Comparison with methodologyComparison with methodology of TUTof TUT 9 Comparison of topics studied Mathematical Statistics (KNITU, RUSSIA) Mathematical Statistics (TUT, Finland) 3. Interval estimation of the distribution parameters. 3. Tests of hypotheses Concept of confidence interval and confidence probability. Interval estimates the expectation and variance. "Statistical hypotheses, Hypothesis testing, One- and two tailed tests, test statistic, P-probabilities, Tests concerning expectations, Tests concerning variances 4. Testing of statistical hypotheses.  4. 2-TESTSχ General algorithm for statistical hypothesis testing. Parametric and nonparametric hypothesis. Valid and critical areas. Errors 1st and 2nd kind. Power of the test. Goodness Pearson, Kol mogorov statistical hypothesis testing on the distribution law. Criteria for testing statistical hypotheses about the parameters of the distribution (Student, Fisher Cochran).  Goodness-of-fit test, test of independence, contingency tables, test for homogeneity   5. NONPARAMETRIC STATISTICS Sign Test, Signed-Rank Test, Mann-Whitney test, Kruskal-Wallis test"
Comparison with methodologyComparison with methodology of TUTof TUT 10 Using computers KNITU, RUSSIA TUT, Finland Modern lecture technology?multimedia Matlab Assignments: handed in, tutorials, or both? Tutorials only Tutorials Third party supporting material online? (Such as Khan Academy) None used None used Is there supportive teaching or a support center available? Support available at 18-hour laboratory works Which tools are used LMS Black Board, Computer tutorial “Introduction to Mathematics Statistics" Moodle, Matlab LMS Black Board: file sharing, course information Moodle: File sharing, course information Computer tutorial: labolatory works. Matlab: Tutorials
SEFI CompetencesSEFI Competences "Statistics and Probability""Statistics and Probability" 11 CORE 0 Probability interpret data presented in the form of line diagrams, bar charts, pie charts Modul 3 interpret data presented in the form of stem and leaf diagrams, box plots, histograms Modul 3 construct line diagrams, bar charts, pie charts, setm and leaf diagrams, box plots, histograms for suitable data sets Modul 3 calculate the mode, median and mean for a set of data items Modul 2 define the terms 'outcome','event' and 'probability' Modul 1 calculate the probability of an event by counting outcomes Modul 1 calculate the probability of the complement of an event Modul 1 calculate the probability of the union of two mutually-exclusive events Modul 1 calculate the probability of the union of two events Modul 1 calculate the probability of the intersection of two independent events Modul 1
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 12 Data Handling calculate the range, inter-quartile range, variance and standard deviation for a set of data items Modul 3 distinguish between a population and a sample know the difference between the characteristic values (moments) of a population and of a sample construct a suitable frequency distribution from a data set calculate realtive frequencies calculate measures of average and dispersion for a grouped set of data understand the effect of grouping on these measures
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 13 Combinatorics evaluate the number of ways of arranging unline objects in a line Modul 1 evaluate the number of ways of arranging objects in a line, where some are alike evaluate the number of ways of arranging unlike objects in a ring evaluate the number of ways of permuting r objects from n unlike objects evaluate the number of combinations of r objects from n unlike objects use the multiplication principle for combinations
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 14 Probability models define a random variable and a discrete probability distribution Modul 1 state the criteria for a binomial model and define its parameters calculate probabilities for a binomial model state the criteria for a Poisson model and define its parameters calculate probabilities for a Poisson model state the expected value and variance for each of these models
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 15 Simple probability interpret probability as a degree of belief Modul 1 understand the distinction between a priori and a posteriori probabilities use a tree diagram to calculate probabilities know what conditional probability is and be able to use it (Bayes' theorem) calculate probabilities for series and parallel connections
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 16 Normal distribution handle probability statements involving continuous random variables Modul 2 convert a problem involving a normal variable to the area part of its density curve relate the general normal distribution to the standardised normal distribution use tables for the standardised normal variable solve problems involving a normal variable using tables
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 17 Sampling define a random sample Modul 3 know what a sampling distribution is understand the term 'mean squared error' of an estimate understand the term 'unbiasedness' of an estimate
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 18 One-dimensional random variables Одномерные случайные величины compare empirical and theoretical distributions Modul 2 apply the exponential distribution to simple problems apply the normal distribution to simple problems apply the Weibull distribution to simple problems apply the gamma distribution to simple problems
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 19 Two-dimensional random variables Двумерные случайные величины understand the concept of a joint distribution Modul 2 understand the terms 'joint desity function', 'marginal distribution functions' define independence of two random variables solve problems involving linear combinations of random variables determine the covariance of two random variables determine the correlation of two random variables
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 20 Small sample statistics realise that the normal distribution is not reliable when used with small samples Modul 3 use tables of the t-distribution solve problems involving small-sample means using the t-distribution use tables of the F-distribution use pooling of variances where appropriate use the method of pairing where appropriate
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 21 Small sample satistics: chi-square tests Небольшие выборки: хи- квадрат тесты use tables for chi-squared distributions Modul 3 decide on the number of degrees of freedom appropriate to a particular problem use the chi-square distribution in tests of independence (contigency tables) use the chi-square distribution in tests of goodness of fit.
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 22 Analysis of variance Дисперсионный анализ set up the information for a one-way analysis of variance interpret the ANOVA table solve a problem using on-way analysis of variance set up the information for a two-way analysis of variance interpret the ANOVA table solve a problem using two-way analysis of variance
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 23 Multiple linear regression and design of experiments Множественная линейная регрессия и планирование эксперимента understand the ideas involved in a multiple regression analysis appreciate the importance of experimental design recognise simple statistical designs
SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 3Level 3 24 Stochastic processes Statistical quality control Reliability Experimental design Queueing theory and discrete simulation Filtering and control Markov processes and renewal theory Statistical inference Multivariate analysis
Thank you for your attention 25

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Probability Theory and Mathematical Statistics

  • 1. PROBABILITY THEORYPROBABILITY THEORY AND MATHEMATICALAND MATHEMATICAL STATISTICSSTATISTICS Svetlana Medvedeva 2 semester - spring semester of the first year of a bachelor's degree Bachelor's program "Computer Science and Engineering"
  • 2. Information about courseInformation about course Number of groups – 6 (80 students) Number of people in group - 11-16. Of these males - 85%, females – 15% Preliminary courses: Algebra and geometry Mathematical analysis Discrete mathematics 2
  • 3. Distribution of working timeDistribution of working time 3
  • 4. Laboratory workshopLaboratory workshop 4 № № themes Name of the laboratory works Labor Input (h.) 1 2.1 Distribution laws of random variables 2 2 3.1 Study of a sample of measurements and build some variation. Exclusion of gross errors of measurement 4 3 3.2 Construction and research of the evaluation function and the probability density of the random variable 4 4 3.3 Building and research of point and interval estimates of mathematical expectation and variance. 4 5 3.4 Statistical hypothesis testing on the distribution of the random variable. Kolmogorov Criterion. Pearson Criterion 4
  • 5. Topics for workshopsTopics for workshops 5 № п/п № theme s Topic name Labor Input (h.) 1 1.1 Actions with events. Classical and geometric definition of probability. Conditional probability. 2 2 1.1 Formulas of addition and multiplication of probabilities 2 3 1.1 Full probabilities and Bayesian formulas. Bernoulli's Formula 2 4 1.2 The laws of distribution of discrete. Actions with tables of distributions. 2 5 1.2 Function and distribution density of continuous random variable 2 6 1.2 Numerical characteristics of random variables 2 7 2.1 Distribution law and numerical characteristics of a system of two random variables The correlation time of the system of two random variables. The correlation coefficient 2 8 2.1 The conditional distribution of a discrete random variable. Conditional mathematical expectation and variance. Independence of continuous random variables. 2 9 2.2 The law of large numbers. Chebyshev's Inequality 2
  • 6. Comparison with methodologyComparison with methodology of TUTof TUT 6 (KNITU, RUSSIA) (TUT, Finland) (TUT, Finland) Selective/mandatory Mandatory Mandatory Mandatory method of teaching Blended Blended Course SEFI level (Core 0, 1, 2 or 3) 0,1,2 0,1,2 0,1,2 Amount of credits 5 4 4 Duration 18 week 7 weeks 7 weeks Lectures 36 28 28 Practice 18 14 14 Laboratory work / tutorials 18 0 0 Homework (% mandatory) 35 39 39 Exam preparation 36 27 27 Exam 2 4 5 Total (number of hours / credits) 180 / 5 108 / 4 108/4 Distribution of study time Probability Theory and Mathematical Statistics
  • 7. Comparison with methodologyComparison with methodology of TUTof TUT 7 Comparison of topics studied Probability Theory (KNITU, RUSSIA) Probability Theory (TUT, Finland) 1. Random events. 1. Concept of Probability and Calculus Basic concepts of probability theory. Events. Experiments.. Elementary events. Random variable, sample space, event, classical probability, addition theorem, multiplication theorem, conditional probability, total probability, Bayes formula, independence of events 2. Random variables. 2. Probability Distributions Discrete random variables. Continuous random variables. Distribution function. Density distribution. Numerical characteristics of random variables. Mathematical expectation, variance, moments, quantile, the significance level. Basic laws of distribution of random variables. Discrete and continuous distributions, density and cumulative distribution functions, variance and standard deviation, binomial, Poisson distribution, normal distribution, t-, F, and khi^2 distributions 3. Systems random variables. 3. Joint Distributions Random vectors and their distribution. Discrete two- dimensional random variable, its distribution law. Continuous two-dimensional random variable, its distribution law. Conditional distributions. Dependent and independent random variables, the correlation Discrete and continuous distributions, marginal distributions, independency, covariance, correlation, principles of statistical testing, Fundamental limit theorem 4. Limit theorems of probability theory.   Random sequences. Types of convergence of random sequences. Characteristics of the law of large numbers. Chebyshev inequality. Chebyshev's theorem, Bernoulli. The central limit theorem.  
  • 8. Comparison with methodologyComparison with methodology of TUTof TUT 8 Comparison of topics studied Mathematical Statistics (KNITU, RUSSIA) Mathematical Statistics (TUT, Finland) 1. Initial statistical analysis of the results of measurements of the random variable. 1. Fundamental sampling distributions and data descriptions Definition of a random sample. Priori and a posteriori sampling. Order statistics. Blunders measurements and methods of their elimination. Building interval statistical series. Polygon and the histogram. Empirical distribution Random Sampling, Some Important Statistics, Data Displays and Graphical Methods, Sampling distributions, Sampling distributions of means, The sampling distribution of the sample variance, The sampling distribution of the sample, t-Distribution, F- distribution 2. 1. Spot Parameter Estimation of distributions of random variables. 2. One- and two-sample estimation Optimality criteria point estimation parameters: consistency, unbiasedness, efficiency, sufficiency. Point estimates of the expectation and variance and their properties. "Point Estimation and Interval Estimation, Single Sample: Estimating the Mean," Prediction intervals, tolerance limits, Two Samples: Estimating the Difference between two means, Paired observations, estimating a proportion, "Single sample: Estimating the variance, Two samples: Estimating the ratio of two variances" 2.2. Interval estimation of the distribution parameters. Concept of confidence interval and confidence probability. Interval estimates the expectation and variance.
  • 9. Comparison with methodologyComparison with methodology of TUTof TUT 9 Comparison of topics studied Mathematical Statistics (KNITU, RUSSIA) Mathematical Statistics (TUT, Finland) 3. Interval estimation of the distribution parameters. 3. Tests of hypotheses Concept of confidence interval and confidence probability. Interval estimates the expectation and variance. "Statistical hypotheses, Hypothesis testing, One- and two tailed tests, test statistic, P-probabilities, Tests concerning expectations, Tests concerning variances 4. Testing of statistical hypotheses.  4. 2-TESTSχ General algorithm for statistical hypothesis testing. Parametric and nonparametric hypothesis. Valid and critical areas. Errors 1st and 2nd kind. Power of the test. Goodness Pearson, Kol mogorov statistical hypothesis testing on the distribution law. Criteria for testing statistical hypotheses about the parameters of the distribution (Student, Fisher Cochran).  Goodness-of-fit test, test of independence, contingency tables, test for homogeneity   5. NONPARAMETRIC STATISTICS Sign Test, Signed-Rank Test, Mann-Whitney test, Kruskal-Wallis test"
  • 10. Comparison with methodologyComparison with methodology of TUTof TUT 10 Using computers KNITU, RUSSIA TUT, Finland Modern lecture technology?multimedia Matlab Assignments: handed in, tutorials, or both? Tutorials only Tutorials Third party supporting material online? (Such as Khan Academy) None used None used Is there supportive teaching or a support center available? Support available at 18-hour laboratory works Which tools are used LMS Black Board, Computer tutorial “Introduction to Mathematics Statistics" Moodle, Matlab LMS Black Board: file sharing, course information Moodle: File sharing, course information Computer tutorial: labolatory works. Matlab: Tutorials
  • 11. SEFI CompetencesSEFI Competences "Statistics and Probability""Statistics and Probability" 11 CORE 0 Probability interpret data presented in the form of line diagrams, bar charts, pie charts Modul 3 interpret data presented in the form of stem and leaf diagrams, box plots, histograms Modul 3 construct line diagrams, bar charts, pie charts, setm and leaf diagrams, box plots, histograms for suitable data sets Modul 3 calculate the mode, median and mean for a set of data items Modul 2 define the terms 'outcome','event' and 'probability' Modul 1 calculate the probability of an event by counting outcomes Modul 1 calculate the probability of the complement of an event Modul 1 calculate the probability of the union of two mutually-exclusive events Modul 1 calculate the probability of the union of two events Modul 1 calculate the probability of the intersection of two independent events Modul 1
  • 12. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 12 Data Handling calculate the range, inter-quartile range, variance and standard deviation for a set of data items Modul 3 distinguish between a population and a sample know the difference between the characteristic values (moments) of a population and of a sample construct a suitable frequency distribution from a data set calculate realtive frequencies calculate measures of average and dispersion for a grouped set of data understand the effect of grouping on these measures
  • 13. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 13 Combinatorics evaluate the number of ways of arranging unline objects in a line Modul 1 evaluate the number of ways of arranging objects in a line, where some are alike evaluate the number of ways of arranging unlike objects in a ring evaluate the number of ways of permuting r objects from n unlike objects evaluate the number of combinations of r objects from n unlike objects use the multiplication principle for combinations
  • 14. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 14 Probability models define a random variable and a discrete probability distribution Modul 1 state the criteria for a binomial model and define its parameters calculate probabilities for a binomial model state the criteria for a Poisson model and define its parameters calculate probabilities for a Poisson model state the expected value and variance for each of these models
  • 15. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 15 Simple probability interpret probability as a degree of belief Modul 1 understand the distinction between a priori and a posteriori probabilities use a tree diagram to calculate probabilities know what conditional probability is and be able to use it (Bayes' theorem) calculate probabilities for series and parallel connections
  • 16. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 16 Normal distribution handle probability statements involving continuous random variables Modul 2 convert a problem involving a normal variable to the area part of its density curve relate the general normal distribution to the standardised normal distribution use tables for the standardised normal variable solve problems involving a normal variable using tables
  • 17. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 1Level 1 17 Sampling define a random sample Modul 3 know what a sampling distribution is understand the term 'mean squared error' of an estimate understand the term 'unbiasedness' of an estimate
  • 18. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 18 One-dimensional random variables Одномерные случайные величины compare empirical and theoretical distributions Modul 2 apply the exponential distribution to simple problems apply the normal distribution to simple problems apply the Weibull distribution to simple problems apply the gamma distribution to simple problems
  • 19. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 19 Two-dimensional random variables Двумерные случайные величины understand the concept of a joint distribution Modul 2 understand the terms 'joint desity function', 'marginal distribution functions' define independence of two random variables solve problems involving linear combinations of random variables determine the covariance of two random variables determine the correlation of two random variables
  • 20. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 20 Small sample statistics realise that the normal distribution is not reliable when used with small samples Modul 3 use tables of the t-distribution solve problems involving small-sample means using the t-distribution use tables of the F-distribution use pooling of variances where appropriate use the method of pairing where appropriate
  • 21. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 21 Small sample satistics: chi-square tests Небольшие выборки: хи- квадрат тесты use tables for chi-squared distributions Modul 3 decide on the number of degrees of freedom appropriate to a particular problem use the chi-square distribution in tests of independence (contigency tables) use the chi-square distribution in tests of goodness of fit.
  • 22. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 22 Analysis of variance Дисперсионный анализ set up the information for a one-way analysis of variance interpret the ANOVA table solve a problem using on-way analysis of variance set up the information for a two-way analysis of variance interpret the ANOVA table solve a problem using two-way analysis of variance
  • 23. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 2Level 2 23 Multiple linear regression and design of experiments Множественная линейная регрессия и планирование эксперимента understand the ideas involved in a multiple regression analysis appreciate the importance of experimental design recognise simple statistical designs
  • 24. SEFI CompetensSEFI Competens " Statistics and Probability “" Statistics and Probability “ Level 3Level 3 24 Stochastic processes Statistical quality control Reliability Experimental design Queueing theory and discrete simulation Filtering and control Markov processes and renewal theory Statistical inference Multivariate analysis
  • 25. Thank you for your attention 25