Probability Space - Definition

Definition

In short, a probability space is a measure space such that the measure of the whole space is equal to one.

The expanded definition is following: a probability space is a triple consisting of:

  • the sample space Ω — an arbitrary non-empty set,
  • the σ-algebra ⊆ 2Ω (also called σ-field) — a set of subsets of Ω, called events, such that:
    • contains the empty set: ,
    • is closed under complements: if A∈, then also (Ω∖A)∈,
    • is closed under countable unions: if Ai∈ for i=1,2,..., then also (∪iAi)∈
      • The corollary from the previous two properties and De Morgan’s law is that is also closed under countable intersections: if Ai∈ for i=1,2,..., then also (∩iAi)∈
  • the probability measure P: → — a function on such that:
    • P is countably additive: if {Ai}⊆ is a countable collection of pairwise disjoint sets, then P(⊔Ai) = ∑P(Ai), where “⊔” denotes the disjoint union,
    • the measure of entire sample space is equal to one: P(Ω) = 1.

Read more about this topic:  Probability Space

Famous quotes containing the word definition:

    The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.
    Jean Baudrillard (b. 1929)

    It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.
    François, Duc De La Rochefoucauld (1613–1680)

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)