Probability and Statistics Guide

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

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
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
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
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

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

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

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

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

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

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

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

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

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.

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

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

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

Probability and Statistics Guide

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (20)

Semelhante a Probability and Statistics Guide

Semelhante a Probability and Statistics Guide (20)

Mais de metamath

Mais de metamath (20)

Último

Último (20)

Probability and Statistics Guide