Probability theory and stochastic processes

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• ConceptProbability space

  A probability space is the mathematical model of a random experiment: a sample space, a σ-algebra of events and a probability measure on the events. The node probability space gives the precise definition.

• ConceptProbability axioms

  The probability axioms of Kolmogorov are the three rules that a probability function $P$ on the events of a sample space $\Omega$ must obey. First, $P(A) \ge 0$ for each event $A$. Second, $P(\Omega) = 1$. Third, for each sequence $A_1, A_2, \dots$ of pairwise disjoint events, $P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n)$. This is countable additivity. A function that obeys these rules is a probability measure.

• ConceptEvent

  An event is a set of outcomes of a random experiment to which we give a probability. The event happens when the outcome of the experiment is in the set. The node event gives the precise definition.

• ConceptSigma-algebra

  A σ-algebra on a set is a family of subsets that contains the whole set and is closed under complements and countable unions. In probability theory, the σ-algebra is the family of events, the sets to which we give a probability. The node sigma-algebra gives the precise definition.

• ConceptPi-system

  A π-system on a set is a nonempty family of subsets that is closed under intersections of two members. It is the kind of generating family used in the uniqueness theorems of measure theory. Two probability measures that agree on a π-system agree on the σ-algebra it generates.

• ConceptRandom variable

  A random variable is a quantity whose value depends on the outcome of a random experiment. Formally it is a measurable function on the sample space of a probability space. The node random variable gives the precise definition.