University of Technology and Humanities in name of K. Pułaskiego w Radomiu.

Department of Computer Science and Mathematics.

Probability and mathematical statistics

E-publication 2017

Ihor Ohirko, Mateusz Grundo, Rafał Leszczyński,


The probability calculus deals with investigating the laws governing random phenomena. Teaches you how you can get information about your chances of getting acquainted with these rights We are interested in events and can predict the course of these phenomena. Probability is a field of mathematics that deals with the study of random phenomena (models of random phenomena) and the laws governing these phenomena. Probability calculates the chance of a certain phenomenon occurring.
Probability probabilities are related to random games (for example, playing dice) and the desire to know the chance of winning. Probability calculus began to shape in the sixteenth century when it began to notice certain regularities in gambling.
He first saw them and attempted to describe the Italian mathematician Geronimo Cardano (1501-1576). The more serious development of the probability calculus occurred in the 17th century thanks to the work P. de Fermat and B. Pascal (French athematicians).
As a mathematician, we think of a Swiss mathematician mathematician Jacob Bernoullie, who developed these issues in the seventeenth century. Probability is based on combinatorics.

Basic concepts of probability.

Random experience – Experience that can be repeated many times in similar or similar conditions and the result can not be unambiguously foreseen. In practice, any activity that may end up with some unpredictable results.

Elementary event – The behavior of a system that we can not foresee with complete certainty, we call random. The measure of „randomness” is probability.
Note: Elementary events must be exclusive – elementary event It does not contain other elementary events. The initial concept of probability is an elementary event and the space of elementary events Ω. Any subset A ⊂ Ω is called a random event

A collection of elementary events – The set of all possible elementary events ω for a random experience – we denote Ω

Random event A – Any subset of all elementary events – the elementary events space Ω. We call a random event or a brief event. Random events are denoted by capital letters A, B, C, ..

Elemental event favoring event A – If A is a random event, then the elementary event ω, such that ω ∈ A, is called the elementary event that favors the event A.

Event certain – A random event that favors all elementary events that form a set of Ω.

Event impossible- Random event, which is not favored by any elementary events belonging to the set Ω.

Event counter to event A– The opposite of event A, we call event A ‚, which is favored by elementary events of space Ω which do not favor Event A.

Power set – The number of elements in a given set A

Statistics – basic concepts

Statistics – the study of research planning, as well as the collection, Organize, present and analyze data, and draw conclusions and make decisions based on them. The word „statistics” is also used to define the data itself and the size of the derivative.

Population – a collection of all the representatives of the examined feature. Example: Demographic research – census. Quality control – collection All devices of the type  roduced by the factory.

Random sample – a representative sample of the whole population, ie one that It reflects all the features and relationships in it.

Example: Random samples are not, for example, surveys among readers of any journal, Among the passersby on the street, the vote of the viewers in the programs. Statistics – the study of research planning, as well as the collection, organization, presentation and analysis of data, and conclusions making decisions based on them. The word „statistics” is also used to define the data itself and the size of the derivative. We say that the sample is simple if the result of selecting one item does not affects the selection of another item. Example: Pulling without returning balls from urn that is filled with finite numbers  hite and black balls, we are dealing with a sample that is not straight.

Uncertainty of measurements

Basic types of scientific experiments and their objectives:

  •   numerical measurement of physical values (parameter definition)
    Example: measuring the speed of light.
  •   Check if the theory is consistent with the data (testing hypotheses)
    Example: test the hypothesis that the speed of light has increased over the past year.
    Example: Suppose we measure the speed of light and compare it
    score with the current value c = 2.998 * 10^8 m / s, obtaining:

Each measurement of the parameter value must include an error estimate to:

  •   You could test new theories,
  •   compare it with the results from other experiments,
  •   Predict the results of other experiments …

Actions on collections

Działania na zbiorach

De Morgan’s Law

Prawa de Morgana

Distribution rights for addition and multiplication:

Prawa rozdzielności dla dodawania i mnożenia

Combinatorial components

If the space of elementary events is finite, then the probability calculation Events that are subsets of this space facilitate the notion and combinatorial assertions. Rule of product: if a certain operation is performed in k steps, where Step 1 can be done on n1 ways, step 2 on n2 ways, …, finally k-th step in nk ways is the number of N ways to do this the activity is:

Reguła iloczynu

We distinguish two types of draws:

  •   no repetition – once drawn element does not return to population,
  •   with repetitions – the drawn element returns to the population before another draw.
    We distinguish two types of ordering:
  •   The order of random elements is significant (variations, permutations).
  •   The order of random elements is not significant (combinations).

Variations with repetitions

We draw k elements of n elements, while the drawn element is returned to the population each time (draw with return). Everyone from
You can choose between the n-ways. This means that the number of k-expression variations with repetitions from the n-element set is:

Wariacje z powtórzeniami2

Variations without repetition

We draw k elements of n elements, while the drawn element does not return to the population (draw without return). The first element can be selected in n ways, the second one only in n-1, the third in n-2, and the k-th one only in n-k + 1 ways. This means that the number of k-word variations without repetition from the n-element set is:

Wariacje bez powtórzeń2

Permutations without and with repetitions

We draw n elements from n elements without returning. First element You can choose on n ways, second only on n-1, third on n-2, and on Last but not least, in 2 ways. This means that the permutation number without repeats of the n-element set is:

Permutacje bez i z powtórzeniami1

If among n elements we have k different elements, the first of which is repeated n1 times, the second n2 times …, k-th nk times (n1 + n2 + … + nk = n), the number of distinct drawings without return, i.e. the number of permutations z The repetitions of an n-elementary set in which the elements repeat n1, n2, …, nk times are:

Permutacje bez i z powtórzeniami2

Combinations without repetition

We draw k elements of n elements, while the drawn element does not return to the population (draw without return). We are not interested in the order of the drawn elements. So we have to deal with k-element subsets of the n-element set. The number of k-element combinations without repetitions from the n-element set is:

Kombinacje bez powtórzeń

Repeated combinations

Let’s consider elements n of different types. Items of the same kind are treated as indistinguishable. Each set of k elements (k ≶ n) when each element belongs to one of these n types is called a k-element combination with repetitions of n types of elements. The number of k-element combinations with repetitions of the n-types is equal to:

Kombinacje z powtórzeniami2

Related Work:

„Probabilistyka, Rachunek prawdopodobieństwa, statystyka matematyczna, procesy stochastyczne” – Edmund Pluciński,Agnieszka Plucińska; Wydawnictwo: PWN

PROWADZĄCY: prof. Ihor Ohirko

STUDENCI III SEM.: Mateusz Grundo, Rafał Leszczyński.



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