Math 2.0

definition

Probability space

Definition

A probability space is a triple (Ω,F,P)(\Omega, \mathcal{F}, P) with these parts. Ω\Omega is a nonempty set, the sample space. F\mathcal{F} is a σ-algebra on Ω\Omega. Its elements are the events. PP is a probability measure on (Ω,F)(\Omega, \mathcal{F}).

Remarks

For one roll of a fair die: Ω={1,…,6}\Omega = \{1, \dots, 6\}, F\mathcal{F} is the power set of Ω\Omega, and P(A)=∣A∣/6P(A) = |A| / 6. A discrete probability space has a countable Ω\Omega, the power set as F\mathcal{F}, and P(A)=∑ω∈Ap(ω)P(A) = \sum_{\omega \in A} p(\omega) for a function p≥0p \ge 0 with total sum 11. The interval [0,1][0, 1] with its Borel σ-algebra and the length of intervals as PP is a probability space. It models a point chosen uniformly at random in [0,1][0, 1].

Explanations

L1 human

A probability space puts three things together: what can happen, which questions we can ask about it, and how likely each answer is. Every problem in probability starts, at least in the background, with one probability space.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What depends on this

164 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Probability_space, Definition

What links here

Used by