Probability theory
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Probability theory is the mathematical study of probability.
Mathematicians think of probabilities as numbers in the closed interval from 0 to 1 assigned to "events" whose occurrence or failure to occur is random. Probabilities are assigned to events according to the probability axioms.
The probability that an event occurs given the known occurrence of an event is the conditional probability of given ; its numerical value is (as long as is nonzero). If the conditional probability of given is the same as the ("unconditional") probability of , then and are said to be independent events. That this relation between and is symmetric may be seen more readily by realizing that it is the same as saying when A and B are independent events.
Two crucial concepts in the theory of probability are those of a random variable and of the probability distribution of a random variable; see those articles for more information.
A somewhat more abstract view of probability
Mathematicians usually take probability theory to be the study of probability spaces and random variables — an approach introduced by Kolmogorov in the 1930s. A probability space is a triple , where
- is a non-empty set, sometimes called the "sample space", each of whose members is thought of as a potential outcome of a random experiment. For example, if 100 voters are to be drawn randomly from among all voters in California and asked whom they will vote for governor, then the set of all sequences of 100 Californian voters would be the sample space Ω.
- is a σ-algebra of subsets of whose members are called "events". For example the set of all sequences of 100 Californian voters in which at least 60 will vote for Schwarzenegger is identified with the "event" that at least 60 of the 100 chosen voters will so vote. To say that is a σ-algebra implies per definition that it contains , that the complement of any event is an event, and that the union of any (finite or countably infinite) sequence of events is an event.
- is a probability measure on , i.e., a measure such that , .
It is important to note that is a function defined on and not on .
If is denumerable we almost always define as the power set of , i.e which is trivially a σ-algebra and the biggest one we can create using . In a discrete space we can therefore omit and just write to define it. If on the other hand is non-denumerable and we use we get into trouble defining our probability measure because is too 'huge', i.e. there will be sets to which it will be impossible to assign a unique measure, e.g the Banach–Tarski paradox. So we have to use a smaller σ-algebra (e.g. the Borel algebra of , which is the smallest σ-algebra that makes all open sets measurable).
A random variable is a measurable function on . For example, the number of voters who will vote for Schwarzenegger in the aforementioned sample of 100 is a random variable.
If is any random variable, the notation , is shorthand for , so that "" is an "event".
For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
Philosophy of application of probability
Some statisticians will assign probabilities only to events that are random, i.e., random variables, that are outcomes of actual or theoretical experiments; those are frequentists. Others assign probabilities to propositions that are uncertain according either to subjective degrees of belief in their truth, or to logically justifiable degrees of belief in their truth. Such persons are Bayesians.
A Bayesian may assign a probability to the proposition that 'there was life on Mars a billion years ago,' since that is uncertain, whereas a frequentist would not assign probabilities to statements at all. A frequentist is actually unable to technically interpret such uses of the probability concept, even though 'probability' is often used in this way in colloquial speech. Frequentists only assign probabilities to outcomes of well defined random experiments, that is, where there is a defined sample space as defined above in the theory section. For another illustration of the differences see the two envelopes problem.
See also
- Glossary of probability and statistics
- list of probability topics
- list of statistical topics
- List of publications in statistics
- Predictive modelling
- Fuzzy measure theory
- probability axioms
- probability distribution
- expected value
- inevitability theorem
- likelihood function
- random variable
- sample space
- variance
- statistical independence
- Notation in probability
- Possibility theory
Bibliography
- Pierre Simon de Laplace (1812) Analytical Theory of Probability
- The first major treatise blending calculus with probability theory, originally in French: Theorie Analytique des Probabilités.
- Andrei Nikolajevich Kolmogorov (1950) Foundations of the Theory of Probability
- The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitrechnung) appeared in 1933.
- Harold Jeffreys (1939) The Theory of Probability
- An empiricist, Bayesian approach to the foundations of probability theory.
- Edward Nelson (1987) Radically Elementary Probability Theory
- Discrete foundations of probability theory, based on nonstandard analysis and internal set theory. downloadable. http://www.math.princeton.edu/~nelson/books.html
- Patrick Billingsley: Probability and Measure, John Wiley and Sons, New York, Toronto, London, 1979.
- Henk Tijms (2004) Understanding Probability
- A lively introduction to probability theory for the beginner, Cambridge Univ. Press.
Major fields of mathematics |
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Logic · Set theory · Algebra (Elementary – Linear – Abstract) · Discrete mathematics · Number theory · Analysis · Geometry · Topology · Applied mathematics · Probability · Statistics · Mathematical physics |
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