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Probability theory is the branch of mathematics concerned with probability , the analysis of random phenomena. [ 1 ] The central objects of probability theory are random variables , stochastic processes , and events : mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted.
Informally, a measure has the property of being monotone in the sense that if A is a subset of B , the measure of A is less than or equal to the measure of B . Furthermore, the measure of the empty set is required to be 0.
In probability theory , a probability space or a probability triple is a mathematical construct that models a real-world process (or "experiment") consisting of states that occur randomly .
In probability theory , the sample space or universal sample space , often denoted S , Ω, or U (for " universe "), of an experiment or random trial is the set of all possible outcomes.
In mathematical analysis , a σ-algebra (also sigma-algebra , σ-field , sigma-field ) is a collection of sets satisfying certain properties.
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity . [ 3 ] The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume ) is that a probability measure must assign 1 to the entire probability space.
In probability theory , a stochastic process ( pronunciation: / s t oʊ ˈ k æ s t ɪ k / ), or sometimes random process ( widely used ) is a collection of random variables ; this is often used to represent the evolution of some random value, or system, over time.