probability theory and mathematical statistics

PROBABILITY THEORYPROBABILITY THEORYAND MATHEMATICAL AND 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 course Information 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 geometryMathematical analysisDiscrete mathematics

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Distribution of working timeDistribution of working time

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Laboratory workshopLaboratory workshop

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№ №

themesName of the laboratory works

Labor Input(h.)

1 2.1 Distribution laws of random variables2

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

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4 3.3 Building and research of point and interval estimates of mathematical expectation and variance.

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5 3.4 Statistical hypothesis testing on the distribution of the random variable.

Kolmogorov Criterion. Pearson Criterion4

Topics for workshopsTopics for workshops

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№ п/п

№theme

sTopic 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 1.2Numerical characteristics of random variables

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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

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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

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(KNITU, RUSSIA) (TUT, Finland) (TUT, Finland)

Selective/mandatory Mandatory Mandatory Mandatorymethod of teaching Blended BlendedCourse SEFI level (Core 0, 1, 2 or 3) 0,1,2 0,1,2 0,1,2Amount of credits 5 4 4Duration 18 week 7 weeks 7 weeksLectures 36 28 28Practice 18 14 14Laboratory work / tutorials 18 0 0Homework (% mandatory) 35 39 39Exam preparation 36 27 27Exam 2 4 5Total (number of hours / credits) 180 / 5 108 / 4 108/4

Distribution of study timeProbability Theory and Mathematical Statistics


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Comparison of topics studiedProbability Theory (KNITU, RUSSIA) Probability Theory (TUT, Finland)1. Random events. 1. Concept of Probability and CalculusBasic 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 DistributionsDiscrete 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 DistributionsRandom 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.


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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.


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Comparison of topics studiedMathematical 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-TESTSχGeneral algorithm for statistical hypothesis testing. Parametric and nonparametric hypothesis. Valid and critical areas. Errors 1st and 2nd kind. Power of the test. Goodness Pearson, Kolmogorov 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 STATISTICSSign Test, Signed-Rank Test, Mann-Whitney test, Kruskal-Wallis test"


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Using computers

KNITU, RUSSIA TUT, FinlandModern lecture technology?multimedia MatlabAssignments: handed in, tutorials, or both? Tutorials only TutorialsThird party supporting material online? (Such as Khan Academy) None used None usedIs 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 Competences SEFI Competences "Statistics and Probability""Statistics and Probability"

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CORE 0 Probabilityinterpret 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 Competens SEFI Competens " Statistics and Probability “ " Statistics and Probability “ Level 1Level 1

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Data Handling calculate the range, inter-quartile range, variance and standard deviation for a set of data items Modul 3distinguish between a population and a sampleknow the difference between the characteristic values (moments) of a population and of a sampleconstruct a suitable frequency distribution from a data setcalculate realtive frequenciescalculate measures of average and dispersion for a grouped set of data understand the effect of grouping on these measures

SEFI Competens SEFI Competens " Statistics and Probability “" Statistics and Probability “Level 1Level 1

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Combinatorics evaluate the number of ways of arranging unline objects in a line Modul 1evaluate the number of ways of arranging objects in a line, where some are alikeevaluate the number of ways of arranging unlike objects in a ringevaluate the number of ways of permuting r objects from n unlike objectsevaluate the number of combinations of r objects from n unlike objectsuse the multiplication principle for combinations


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Probability models

define a random variable and a discrete probability distribution

Modul 1

state the criteria for a binomial model and define its parameterscalculate probabilities for a binomial modelstate the criteria for a Poisson model and define its parameterscalculate probabilities for a Poisson modelstate the expected value and variance for each of these models


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Simple probability interpret probability as a degree of belief Modul 1understand the distinction between a priori and a posteriori probabilitiesuse a tree diagram to calculate probabilitiesknow what conditional probability is and be able to use it (Bayes' theorem)calculate probabilities for series and parallel connections


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Normal distribution

handle probability statements involving continuous random variables Modul 2convert a problem involving a normal variable to the area part of its density curverelate the general normal distribution to the standardised normal distributionuse tables for the standardised normal variablesolve problems involving a normal variable using tables


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Sampling define a random sample Modul 3know what a sampling distribution is

understand the term 'mean squared error' of an estimateunderstand the term 'unbiasedness' of an estimate


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One-dimensional random variables

Одномерные случайные величины

compare empirical and theoretical distributions Modul 2apply the exponential distribution to simple problemsapply the normal distribution to simple problemsapply the Weibull distribution to simple problemsapply the gamma distribution to simple problems


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Two-dimensional random variables

Двумерные случайные величины

understand the concept of a joint distribution Modul 2understand the terms 'joint desity function', 'marginal distribution functions'define independence of two random variablessolve problems involving linear combinations of random variablesdetermine the covariance of two random variablesdetermine the correlation of two random variables


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Small sample statistics

realise that the normal distribution is not reliable when used with small samples Modul 3use tables of the t-distributionsolve problems involving small-sample means using the t-distributionuse tables of the F-distributionuse pooling of variances where appropriateuse the method of pairing where appropriate


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Small sample satistics: chi-square

testsНебольшие

выборки: хи-квадрат тесты

use tables for chi-squared distributions Modul 3decide on the number of degrees of freedom appropriate to a particular problemuse the chi-square distribution in tests of independence (contigency tables)use the chi-square distribution in tests of goodness of fit.


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Analysis of varianceДисперсионный

анализ

set up the information for a one-way analysis of varianceinterpret the ANOVA tablesolve a problem using on-way analysis of varianceset up the information for a two-way analysis of varianceinterpret the ANOVA tablesolve a problem using two-way analysis of variance


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Multiple linear regression and

design of experiments

Множественная линейная

регрессия и планирование эксперимента

understand the ideas involved in a multiple regression analysisappreciate the importance of experimental designrecognise simple statistical designs


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Stochastic processes

Statistical quality controlReliabilityExperimental design

Queueing theory and discrete simulation

Filtering and controlMarkov processes and renewal theory

Statistical inferenceMultivariate analysis

Thank you for your attention

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probability theory and mathematical statistics

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