Probability theory

Probability theory and probability theory is a branch of mathematics that emerged from the formalization of modeling and the study of random events. Together with the mathematical statistics, which tells about the underlying model based on observations of random processes, it forms the mathematical subfield of stochastics. The central objects of probability theory are random events, random variables and stochastic processes.

• 2.1 Laplace experiments
• 2.2 Conditional Probability 2.2.1 joint probability ( intersection of events)
• 2.2.2 Bayes' Theorem

• 3.1 Probability Spaces
• 3.2 random variable

Axiomatic structure

Like every branch of modern mathematics is also formulated the theory of probability and set theory based on axiomatic specifications. Starting point of the probability theory are events that are regarded as quantities and with which are associated probabilities; Probabilities are real numbers between 0 and 1; the assignment of probabilities to events must meet certain minimum requirements.

These definitions do not indicate how to determine the probabilities of individual events; they also say nothing about what chance and what probability really are. The mathematical formulation of probability theory is thus open to various interpretations, their results are still accurate and the respective understanding of the concept of probability independent.

Definitions

Conceptually, it is considered as the basis of the mathematical examination of a random process or random experiment. All possible outcomes of this random process is summarized in the result set. Frequently, however, not interested in the exact result, but only as to whether it is in a certain subset of the result set or not, which can be interpreted to mean that an event has occurred or not. Thus, an event is defined as a subset of the. Contains the event exactly one element of the result set, it is a natural event. Composite events contain multiple results. So the result is an element of the result set, the event is a subset.

So you can assign probabilities to the events in a meaningful way, they will be listed in a lot of system algebra of events or event space above, a set of subsets of, for which: contains and is a body, that is, they is over the set operations of union and complementation ( with respect to relative ) completed as well as towards the infinite union of countably many sets. The probabilities are then images of a certain picture of the event space in the interval [ 0,1]. Such a mapping is called probability. The triple is called a probability space.

Axioms of Kolmogorov

The axiomatic justification of probability theory was developed in the 1930s by Andrei Kolmogorov. A probability measure must therefore satisfy the following three axioms:

Example: As part of a physical modeling a probability to describe the outcome of a coin toss is set (called events) the possible outcomes may be the number and head.

• Then the result set.
• As an event space, the power set can be selected, ie.
• For the probability measure is due to the axioms that:

Additional physical assumptions about the nature of the coin can now run at about the choice.

Conclusions

From the axioms arise directly some conclusions:

1 From the additivity of the probability of disjoint events follows that complementary events (mating events) complementary probabilities (counter probabilities ) have: .

2 It follows that the impossible event, the empty set, the probability is zero.

3 For the union of disjoint events is not necessary follows:.

Furthermore, a distinction between countable and uncountable result sets.

Countable result set

For a countable result set each elementary event can be a positive probability to be assigned. If finite or countably infinite, one can for the σ - algebra choose the power set of. The sum of the probabilities of all the elementary events is here 1

Uncountable result set

A prototype of an uncountable result set is the set of real numbers. In many models, it is not possible to make sense to assign all subsets of the real numbers a probability. An event system is chosen instead of the power set of the real numbers here mostly the Borel σ -algebra is the smallest σ - algebra which contains all the intervals of real numbers as elements. The elements of this σ - algebra are called Borel sets or ( Borel ) measurable. If the probability of each Borel set as integral

Can be written from a probability density, is called absolutely continuous. In this case ( but not only in this ) have all elementary events { x } is the probability 0, the probability density of an absolutely continuous probability measure is only almost everywhere uniquely determined, ie they can from on any Lebesgue null set, ie a set Lebesgue measure 0, changed, without being changed. If the first derivative of the distribution function of exist, it is a probability density of P. The values ​​of the probability density are not interpreted as probabilities.

Special features in the case of discrete probability spaces

Laplace experiments

If one assumes that only a finite number of elementary events and all are equally possible, that is, with the same probability occur (such as tossing a coin ideal where { number } and { head }, respectively, the probability possess 0.5 ) so it is called a Laplace experiment. Then, probabilities can be calculated easily: We take a finite amount of result, which has the cardinality, that is, they have elements. Then the probability of each elementary event is simple.

As a consequence it follows that the corresponding multiple probability applies to events that are composed of several elementary events. If an event is the thickness, as is the association of elementary events. Each of them has a probability, that is. One thus gets the simple relation

Wherein the Laplacian probability of an event trials is equal to the number of the favorable results of this event, divided by the total number of possible results.

The picture below shows an example of the dice with an ideal cube.

A typical Laplace test is also drawing a card from a game with cards or drawing a ball from an urn with balls. Here each elementary event has the same probability. To determine the number of elementary events in Laplace experiments, methods of combinatorial analysis are used frequently.

The concept of Laplace experiments can be generalized to the case of a continuous uniform distribution.

Conditional Probability

Under a conditional probability refers to the probability of the occurrence of an event on condition that the occurrence of another event is already known. Course must be able to enter, so it should not be the impossible event. It then writes or rare for "Probability of assuming " short " of provided ".

Example: The probability of drawing a heart card from a Skatblatt ( event) is 1/4 because there are 32 cards and including 8 heart cards. Then is. The event counter is then diamonds, spades or clubs and therefore has the chance.

If, however, already the event " The card is red " has occurred ( it was drawn a heart or check card, but it is not known which of the two colors), one thus has only the choice among the 16 red cards, then the probability that it is then the heart sheet.

This consideration was for a Laplaceversuch. The general case is defined by the conditional probability "provided " as

The fact that this definition makes sense, reflected in the fact that with this probability the axioms of Kolmogorov is enough to to be limited as a new result set; That is, the following applies:

Proof:

Example: It is above the event " drawing a heart card" and the event " It's a red card ." Then:

And

Consequently, the following applies:

From the definition of conditional probability has the following consequences:

Joint probability ( intersection of events)

The simultaneous occurrence of two events and corresponding set-theoretically the occurrence of the composite event. The probability thereof is calculated for the joint probability or joint probability

Proof: By the definition of conditional probability on the one hand

And on the other hand

Changing after then instantly the assertion.

Example: It is pulled a card from 32 cards. be the event: "It's a king." be the event: "It's a heart card ". Then the simultaneous occurrence of and so the event: " The card drawn is a heart - King". Is apparent. Further, because there is only one heart card among the four kings. And, in fact, is the probability for the King of hearts.

Bayes' Theorem

The conditional probability that the condition can be achieved by the conditional probability under the condition

Express, if the total probabilities P (B ) and P ( A) knows ( Bayes theorem).

Dependence and independence of events

Events called independently when the occurrence of a probability of the other non- influenced. In the opposite case they are called dependent. One defines:

That this is the term " independence" justice can be seen by rearranging by:

This means that the total probability is as large as the probability for granted; the occurrence of affected so the chance of not.

Example: There is a drawn out 32 cards. be the event "It's a heart card ". be the event "It's an image - map ". These events are independent, because the knowledge that you draw a picture card, does not affect the likelihood that it's a heart card (the percentage of images among heart cards is as great as the proportion of the images on all cards ). Is evidently and. is the event "It's a heart - Picture Card". Since there are three of them, is. And in fact, it is found that is.

Another example of very small and very large probabilities can be found in Infinite -Monkey Theorem.

Measure-theoretical point of view

The classical probability theory considers only probabilities on discrete probability spaces and continuous models with density functions. These two approaches can unify and generalize the modern formulation of probability theory, integration theory is based on the concepts and results of the measurements and.

Probability spaces

In this view, a probability space is a measure space with a probability measure. This means that the result set is any set, the event space is a σ - algebra with base set and is a measure that is normalized by.

Important cases of default probability spaces are:

• Is a countable set and the power set of. Then each probability measure is uniquely determined by its values ​​on the one-element subsets of and applies to all

• Is a subset of, and σ is the Borel - algebra. If the probability measure absolutely continuous with respect to Lebesgue measure, then have by the theorem of Radon - Nikodym a Lebesgue density, ie, valid for all

• Is a Cartesian product and is the product σ - algebra of σ - algebras. Are given probability measures on, then by the product measure defined on a probability measure, which models the independent sequential execution of individual experiments.

Random variable

A random variable is the mathematical concept of a quantity whose value depends on luck. From maßtheoretischer point of view it is a measurable function on a probability space in a measuring space consisting of a set and a σ - algebra. Measurability means that for all the archetype is an element of the σ - algebra. The distribution is then nothing more than the size

Which is induced on the measuring of space, and this makes it a probability space.

The expected value of a real-valued random variable averages the possible outcomes. It can be abstractly defined as the integral of with respect to the probability measure:

Probability Theory and Statistics

Probability theory and mathematical statistics are collectively referred to as stochastic. Both areas are closely interrelated:

• Statistical distributions are regularly modeled under the assumption that they are the result of random processes.
• Statistical methods can provide clues to numerical way for the behavior of probability distributions.

Areas of application

Probability theory arose from the problem of equitable distribution of the insert with broken gambling. Other early applications are in the area of gambling.

Today, probability theory is a basis of mathematical statistics. Statistics Applied uses results of probability theory to analyze survey results or to create economic forecasts.

Large areas of physics such as thermodynamics and quantum mechanics use the probability theory for the theoretical description of their results.

It is also the basis for mathematical disciplines such as Reliability Theory, renewal theory and queuing theory and the tool for analysis in these areas.

In the pattern recognition, the probability theory of central importance.