Probability Spaces

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本文介绍"Probability Spaces", 这是概率论中的基本概念. 注意到, 概率论的公理化基础是测度论, 因此对于概率度量函数\(\mathbb P\)的讨论涉及一些测度论知识.

Ref:

  1. NOTES ON PROBABILITY by Greg Lawler
  2. Probability_Theory by Kyle Siegrist
  3. The Book of Statistical Proofs

Random Experiment

Definition: A random experiment is any repeatable procedure that results in one out of a well-defined set of possible outcomes.

  • The set of possible outcomes is called sample space.
  • A set of zero or more outcomes is called a random event.
  • A function that maps from events to probabilities is called a probility measure

Together, sample space, event space and probility measure characterize a random experiment.

Probability Space

Definition: A probability space is a measure space with total measure one. The standard notation is \((\Omega, \mathcal F, \mathbb P)\) where:

  • the sample space \(\Omega\).
  • an event space \(\mathcal{F} \subseteq 2^\Omega\).
  • a probability measure \(\mathbb P: \; \mathcal{F} \rightarrow [0,1]\), i.e. a function mapping from the event space to the real numbers, observing the axioms of probability.

\(\Omega\)

Definition: \(\Omega\) is a set of all possible outcomes, which are denoted \(\omega\) , from this experiment.

  • \(\Omega\) is sometimes called a sample space in elementary probability.

\(\mathcal F\)

Definition: \(\mathcal F\) is a \(\sigma\)-algebra (or \(\sigma\)-field, we will use these terms synonymously) of subsets of \(\Omega\) . Sets in \(\mathcal F\) are called events.

  • \(\mathcal F\) is sometimes called a event space in elementary probability.
  • 简单地说, \(\mathcal F\) is a set of events.
  • Every event \(E\) is a ertain subset of \(\Omega\). 每个事件(event)都是若干个outcome的集合, 也就是\(\Omega\)的一个子集.

Event

对于某个event \(E\), 它本身只是若干outcome的集合, 也就是说\(E \subseteq \Omega\), \(E \in \mathcal F\), 但我们用如下"statement"来指代(或者说定义)它:

Suppose that \(E \subseteq \mathcal F\) is a given event, and that the experiment is run, resulting in outcome \(\omega \subseteq \Omega\),

  1. If \(\omega \in E\) then we say that \(E\) occurs.
  2. If \(\omega \notin E\) then we say that \(E\) does not occur.

\(E\)本身其实让statement成立所需的outcome, 即让indicator function \[ 1_E(\omega)= \begin{cases}1, & \omega \in E, \\ 0, & \omega \notin E .\end{cases} \] \(1_E(\omega)= 1\)成立的值, 其形式是\(\omega\)的集合 \(w_i,w_j,w_k, ...\)., 但我们用它来指代\(1_E(\omega)\)=1, 也就是"statement成立"这个事件.

例如: 掷一次骰子的结果有六种情况, \(\omega = \{ 1,2,3,4,5,6\}\), 当我们说出statement:"骰子的结果是奇数" 时, 我们实际上就定义了一个event \(A = \{ 1,3,5\}\). 它本身是让\(1_{骰子的结果是奇数}(\omega)= 1\)成立的值\(\{1,3,5\}\), 但我们用它来指代"骰子的结果是奇数成立"这一事件.

\(\mathbb P\)

A probability measure (or probability distribution) \(\mathbb P\) is a function from \(\mathcal F\) to \([0, 1]\) that satisifes the following axioms:

  1. \(\mathbb P(E) \ge 0\) for every event \(E\).

  2. \(\mathbb P(\Omega) = 1\)

  3. If events \(E_1, E_2, . . . \in \mathcal F\) are disjoint, \[ \mathbb P \left(\bigcup_{i}^\infty E_i\right) = \sum_{i}^\infty \mathbb P(E_i) \] We say "probability of \(E\)" for \(\mathbb P(E)\).

\(\mathbb P(E)\) is a measure of the likelihood of event \(E\) to occur.

\(\mathbb P(E)\) is often written as \(p(E)\), \(P(E)\) or \(\text{Pr}(E)\).

Discrete Probability Space

A discrete probability space is a probability space such that \(\Omega\) is finite or countably infinite. In this case we usually choose \(\mathcal F\) to be all the subsets of \(\Omega\) (this can be written \(\mathcal F = 2^\Omega\)1)

The probability measure \(\mathbb P\) is given by a function \[ p : \Omega → [0, 1] \] with \(\sum_{\omega \in \Omega} p(\omega) = 1\).

Definition: Probability is a measure of the likelihood of an event to occur.

Note: \(\mathbb P(E)\) can also be written as \(p(A)\), \(P(A)\) or \(\text{Pr}(A)\).

  1. \(2^\Omega\)是 \(\Omega\) 的幂集(Power Set).↩︎