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## Probability theory
Mathematicians think of probabilities as numbers in the 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 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
"Pure" mathematicians usually take probability theory to be the study of probability spaces and random variables — an approach introduced by Andrey Nikolaevich Kolmogorov in the 1930s. A probability space is a triple (Ω,
- Ω 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 for governor, then the set of all sequences of 100 Californian voters would be the sample space Ω.
*F*is a sigma-algebra of subsets of Ω 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*F*is a sigma-algebra necessarily implies that the complement of any event is an event, and the union of any (finite or countably infinite) sequence of events is an event.- P is a probability measure on
*F*, i.e., a measure such that P(Ω) = 1.
P is defined on F and not on Ω.
With Ω denumerable we can define F := powerset(Ω) which is trivially a sigma-algebra and the biggest one we can create using Ω.
In a discrete space we can therefore omit F and just write (Ω, P) to define it. If on the other hand Ω is non-denumerable and we use F = powerset(Ω) we get into trouble defining our probability measure P because F is too 'huge'. So we have to use a smaller sigma-algebra F. We call this sort of probability space a continuous probability space and are lead to questions in measure theory when we try to define P.A random variable is a measureable function on Ω. For example, the number of voters who will vote for Schwarzenegger in the aforementioned sample of 100 is a random variable.
If ## Philosophy of application of probability
Some statisticians will assign probabilities only to events that they think of as random, according to their relative frequencies of occurrence, or to subsets of populations as proportions of the whole; those are See also: probability, probability axioms, probability distribution, random variable, statistical independence, likelihood, expectation, variance ## External Links: | |||||||

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