Stochastics

The Stochastic (from ancient Greek στοχαστικὴ τέχνη stochastikē Techne , Latin ars conjectandi 'art of conjecture , rate art') is a branch of mathematics and holds a generic term which areas Probability theory and mathematical statistics together. Events or results are referred to as stochastic which do not always occur when the same process is repeated, sometimes only sometimes and whose occurrence cannot be predicted for the individual case.

The historical aspects are presented in the article History of the Calculus of Probability .

overview

Mathematical stochastics deals with the description and investigation of random experiments such as the throwing of dice or coins, as well as temporal developments and spatial structures influenced by chance .

Such events, developments and structures are often documented by data, for the analysis of which the statistics provide suitable methods. In this case, the random influences usually arise in the context of the random selection of a sample from an actually interesting population .

Overall, stochastics thus includes a spectrum of methods with which one can determine both the probability of winning the lottery or the size of the uncertainty in opinion polls . Stochastics is also important for financial mathematics and its methodology helps, for example, with pricing options .

Probabilities and Random Experiments

A forecast is understood to mean:

Specification of probabilities

Probabilities are represented with the letter (from French probabilité , introduced by Laplace ) or . They do not have a unit, but are numbers between zero and one , with zero and one also being permissible probabilities. Therefore, they can be given as percentages (20%), decimal numbers ( ), fractions ( ), odds (2 of 10 or 1 of 5) or ratios (1 to 4) (all details describe the same probability). ${\ displaystyle \ P}$${\ displaystyle \ W}$${\ displaystyle 0 {,} 2}$${\ displaystyle {\ tfrac {2} {10}}}$

Often misunderstandings arise if the correct distinction is not made between “to” and “from”: “1 to 4” means that the one desired event is offset by 4 undesired events. Thus, there are 5 events of one which is what you are, so "1 of 5".

If a random experiment is carried out several times in a row, the relative frequency of an event can be calculated by dividing the absolute frequency , i.e. the number of successful attempts, by the number of attempts made. For an infinite number of attempts, this relative frequency is converted into the probability. In practice, the number of attempts required to achieve an acceptable match between relative frequency and probability is often underestimated.

Probabilities zero and one - impossible and certain events

The fact that an event is assigned the probability of zero only means that its occurrence is in principle impossible if there are only a finite number of different test outcomes.

This is illustrated by the following example: In a random experiment, any real number between 0 and 1 is drawn. It is assumed that every number is equally probable - that is, the uniform distribution over the interval is assumed. Then, since there are infinitely many numbers in the interval, the probability of occurrence for each individual number from the interval is zero, but any number from is possible as a result of the drawing. ${\ displaystyle [0,1]}$${\ displaystyle [0,1]}$

In the context of this example, an impossible event is the drawing of 2, i.e. the elementary event . ${\ displaystyle \ {2 \}}$

An event is said to be “certain” if it has a probability of 1. The probability that an impossible event will not occur is 1 and it is a certain event. An example of a certain event when rolling a six-sided die is the event "no seven is rolled".

Constraints of integrity, system of axioms

Basic assumptions of stochastics are described in the Kolmogorov axioms according to Andrei Kolmogorov . From these and their implications it can be concluded that:

The probability of the event, which includes all possible test outcomes, is : ${\ displaystyle 1}$

${\ displaystyle \ P (\ Omega) = 1.}$

The probability of an impossible event is : ${\ displaystyle 0}$

${\ displaystyle P (\ emptyset) = 0.}$

All probabilities are between zero and one, inclusive:

${\ displaystyle 0 \ leq P (A) \ leq 1.}$

The probability of an event occurring and that of not occurring add up to one:

${\ displaystyle P (A) + P ({\ bar {A}}) = 1.}$

In a complete system of events (for this they must all be pairwise disjoint and their union set must be the same ) the sum of the probabilities is equal to : ${\ displaystyle A_ {i}}$${\ displaystyle A_ {i}}$ ${\ displaystyle \ Omega}$${\ displaystyle 1}$

${\ displaystyle \ sum _ {i = 1} ^ {n} P (A_ {i}) = 1.}$

Laplace experiments

Laplace experiments, named after the mathematician Pierre-Simon Laplace , are random experiments for which the following two points are met:

• There are only a finite number of possible test outcomes.
• All possible outcomes are equally likely.

Simple examples of Laplace experiments are tossing the ideal dice, tossing a coin (if you ignore the fact that it can stay on the edge) and drawing the lottery numbers.

The probability of a Laplace experiment is calculated according to ${\ displaystyle P}$

${\ displaystyle P (E) = {\ frac {\ text {Number of test outputs that are favorable for the result}} {\ text {Number of possible test outputs}}} \,}$

Combinatorics

Combinatorics is a branch of mathematics that deals with determining the number of possible arrangements or selections of

• distinguishable or indistinguishable objects
• with or without observing the order

employed. In modern combinatorics, these problems are reformulated as mappings, so that the task of combinatorics can essentially be limited to counting these mappings.

Game theory

Game theory is a branch of mathematics that deals with analyzing systems with several actors (players, agents). Game theory tries, among other things, to derive the rational decision-making behavior in social conflict situations. The stochastics come into play at various points. First, in games like Battle of the Sexes , where the best possible strategy is to make a decision randomly. On the other hand, game theory also deals with the systems in which the actors do not know the complete situation, i.e. they do not have complete information . Then they have to choose an optimal game strategy based on their guesses.

statistics

Statistics is a mathematics- based methodology for analyzing quantitative data. She combines empirical data with theoretical models.