- •Introduction
- •Basic concepts of probability theory
- •Classical definition of probability
- •Relative frequency
- •Geometric probabilities
- •Glossary
- •Exercises for Seminar 1
- •Exercises for Homework 1
- •Basic formulas of combinatorial analysis
- •Operations over events
- •Glossary
- •Exercises for Seminar 2
- •Exercises for Homework 2
- •Theorem of addition of probabilities of incompatible events
- •Complete group of events
- •Opposite events
- •Conditional probability
- •Theorem of multiplication of probabilities
- •Glossary
- •Exercises for Seminar 3
- •Exercises for Homework 3
- •Independent events
- •Where a is the appearance of at least one of the events a1, a2, …, An; .
- •Glossary
- •Exercises for Seminar 4
- •Exercises for Homework 4
- •Theorem of addition of probabilities of compatible events
- •Formula of total probability
- •Probability of hypotheses. Bayes’s formulas.
- •Glossary
- •Exercises for Seminar 5
- •Exercises for Homework 5
- •Repetition (recurrence) of trials. The Bernoulli formula
- •Local theorem of Laplace
- •Integral theorem of Laplace
- •Glossary
- •Exercises for Seminar 6
- •Exercises for Homework 6
- •Random variables. The law of distribution of a discrete random variable
- •A random variable is understood as a variable which as result of a trial takes one of the possible set of its values (which namely – it is not beforehand known).
- •Mathematical operations over random variables
- •(Mathematical) expectation of a discrete random variable
- •Dispersion of a discrete random variable
- •Glossary
- •Exercises for Seminar 7
- •Exercises for Homework 7
- •Distribution function of a random variable
- •Properties of a distribution function
- •Continuous random variables. Probability density
- •Properties of probability density
- •Glossary
- •Exercises for Seminar 8
- •Exercises for Homework 8
- •Basic laws of distribution of discrete random variables
- •1. Binomial law of distribution
- •2. The law of distribution of Poisson
- •3. Geometric distribution
- •4. Hypergeometric distribution
- •Glossary
- •Exercises for Seminar 9
- •Exercises for Homework 9
- •Basic laws of distribution of continuous random variables
- •1. The uniform law of distribution
- •2. Exponential law of distribution
- •3. Normal law of distribution
- •Glossary
- •Exercises for Seminar 10
- •Exercises for Homework 10
- •The law of large numbers and limit theorems
- •The central limit theorem
- •Glossary
- •Exercises for Seminar 11
- •Exercises for Homework 11
- •Mathematical statistics. Variation series and their characteristics
- •Numerical characteristics of variation series
- •Glossary
- •Exercises for Seminar 12
- •Exercises for Homework 12
- •Bases of the mathematical theory of sampling
- •Glossary
- •Exercises for Seminar 13
- •Exercises for Homework 13
- •Methods of finding of estimations
- •Notion of interval estimation
- •Glossary
- •Exercises for Seminar 14
- •Exercises for Homework 14
- •Testing of statistical hypotheses
- •Glossary
- •Exercises for Seminar 15
- •Exercises for Homework 15
- •Individual homeworks
- •Variant 1
- •Variant 2
- •Variant 3
- •Variant 4
- •Variant 5
- •Variant 6
- •Variant 7
- •Variant 8
- •Variant 9
- •Variant 10
- •Variant 11
- •Variant 12
- •Variant 13
- •Variant 14
- •Variant 15
- •Variant 16
- •Variant 17
- •Variant 18
- •Variant 19
- •Variant 20
- •Variant 21
- •Variant 22
- •Variant 23
- •Variant 24
- •Variant 25
- •Final exam trial tests (for self-checking)
- •Appendix
- •Values the functions and
- •List of the used books
- •Contents
Introduction
Events (phenomena) observed by us can be subdivided on the following three kinds: reliable, impossible and random.
A reliable (universal) event is an event that necessarily will happen if a certain set of conditions S holds. For example, if a vessel contains water with a normal atmospheric pressure and temperature 20 degrees, the event «water in a vessel is in a liquid state» is reliable. In this example the given atmospheric pressure and temperature of water make the set of conditions S.
An impossible (null) event is an event that certainly will not happen if the set of conditions S holds. For example, the event «water in a vessel is in a rigid state» certainly will not happen if the set of conditions of the previous example holds.
A random event is an event that can either take place, or not to take place for holding the set of conditions S. For example, if a coin is tossed it can land on one of two sides: heads or tails. Therefore, the event «the coin lands on heads» is random. Each random event, in particular an appearance of heads, is the consequence of a functioning very many random reasons (in our example: the force with which the coin is tossed, the form of the coin and many others). It is impossible to take into account an influence on result of all these reasons as their number is very great and laws of their functioning are unknown. Therefore probability theory does not pose the problem to predict that a single event whether or not will happen – it simply is unable to solve this problem.
We have another picture if we consider random events that can multiply be observed for holding the same conditions S, i.e. if the speech goes on mass homogeneous random events. It appears that a rather large number of homogeneous random events independently from their concrete nature are subordinated to definite regularities, namely probability regularities. Probability theory studies these regularities. Thus, the subject of probability theory is studying probability regularities of mass homogeneous random events.
Methods of probability theory are widely applied in various branches of natural sciences and techniques: theory of reliability, theory of mass service, theoretical physics, geodesy, astronomy, theory of shooting, theory of mistakes of observation, theory of automatic control, general theory of communication and many others theoretical and applied sciences. Probability theory serves also for a substantiation of mathematical statistics which is in turn used at planning and organizing a manufacture, at analysis of technological processes, and for many other purposes.
All life a person has to make decisions: in personal sphere (in which university or college to enter, with whom to communicate; how to study); in public (to attend evenings, theatres, meetings, assemblies, elections); in industrial (determining factors essentially influencing on productivity, quality of materials and etc.); in scientific (promotion and checking scientific hypotheses).
Decision-making usually pursues one of the following purposes: forecasting of a future state of a process (an object); control (i.e. how should to change certain parameters of an object in order that other parameters have taken on a desirable value); an explanation of internal structure of an object.
Usually decision-making is preceded with an analysis of known data (on the basis of previous experience, common sense, intuition, and etc.). Aspiring to see and prove regularities in uncertain processes, the mankind has developed the whole arsenal of methods which refer to as mathematical statistics (applied statistics or data analysis).
Mathematical statistics is a section of mathematics in which mathematical methods of ordering, processing and using statistical data for scientific and practical conclusions are studied.
Abraham Wald (1902-1950) spoke that «mathematical statistics is theory of decision-making in conditions of uncertainty».
In essence mathematical statistics gives a unique, mathematically proved apparatus for solving problems of control and forecasting at absence of obvious regularities (presence of randomness) in investigated processes.
It is wonderful that the science begun with consideration of gambles was fated to become the major object of human knowledge …
Pierre-Simon Laplace (1749-1827)
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