Probability theory

Probability theory is the mathematical study of probability.

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 Kolmogorov in the 1930s. A probability space is a triple (Ω, F, P), where

Related Topics:
Kolmogorov - 1930s - Probability space

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• 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 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.

It is important to note that P is defined on F and not on Ω.

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With Ω denumerable we can define F := powerset(Ω) which is trivially a sigma-algebra and the biggest one we can create using Ω.

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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 (e.g. the Borel algebra of Ω). We call this sort of probability space a continuous probability space and are led to questions in measure theory when we try to define P.

Related Topics:
Non-denumerable - Borel algebra - Measure theory

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A random variable is a measurable function on Ω. For example, the number of voters who will vote for Schwarzenegger in the aforementioned sample of 100 is a random variable.

Related Topics:
Random variable - Measurable function

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If X is any random variable, the notation P(X ≥ 60) is shorthand for P({ ω in Ω : X(ω) ≥ 60 }), so that "X ≥ 60" is an "event".

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For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.

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