Probability Spaces
Sources:
- NOTES ON PROBABILITY by Greg Lawler
Probability Spaces
Notation
| Symbol | Type | Description |
|---|---|---|
| $ $ | Set | Sample space, the set of all possible outcomes |
| $ $ | Element of $ $ | A specific outcome in the sample space |
| $ $ | \(\sigma\)-algebra | Event space, the collection of subsets of $ $ that satisfy the properties of a \(\sigma\)-algebra |
| $ $ | Function $ : $ | Probability measure, a function $ : $ satisfying Kolmogorov’s axioms |
| $ X $ | Random variable | A measurable function $ X : $ |
| $ _X $ | Function $ _X : $ | Distribution of the random variable $ X $, defined on Borel subsets of $ $ |
| $ $ | \(\sigma\)-algebra | The Borel \(\sigma\)-algebra on $ $ |
| $ F_X(x) $ | Cumulative distribution function (CDF) | Probability that $ X $ takes a value less than or equal to $ x $, $ F_X(x) = (X x) $ |
| $ f_X(x) $ | Probability density function (PDF) | Describes the density of $ X $ if $ X $ is absolutely continuous |
| $ p_X(x) $ | Probability mass function (PMF) | Describes the probability of $ X $ taking a specific value $ x $ if $ X $ is discrete |
| $ $ | Set | The set of real numbers |
| $ (, , _X) $ | Probability space | The transformed probability space induced by the random variable $ X $ |
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.
Definition
A probability space is a measure space with total measure one. The standard notation is \((\Omega, \mathcal F, \mathbb P)\) where:
\(\Omega\) is a set (sometimes called a sample space in elementary probability). Elements of \(\Omega\) are denoted \(\omega\) and are sometimes called outcomes.
\(\mathcal{F}\) is a \(\sigma\)-algebra (or \(\sigma\)-field, we will use these terms synonymously) of subsets of \(\Omega\). Sets in \(\mathcal{F}\) are called events.
\(\mathbb{P}\), the probility measure, is a function from \(\mathcal{F}\) to \([0,1]\) with \(\mathbb{P}(\Omega)=1\) and such that if events \(E_1, E_2, \ldots \in \mathcal{F}\) are disjoint, \[ \mathbb{P}\left[\bigcup_{j=1}^{\infty} E_j\right]=\sum_{j=1}^{\infty} \mathbb{P}\left[E_j\right] \] We say "probability of \(E\)" for \(\mathbb{P}(E)\).
Event and indicator function
The indicator function, denoted \(1_E(\omega)\), is a mathematical tool used to signify whether a specific outcome \(\omega\) from the sample space \(\Omega\) belongs to an event \(E\). It is formally defined as:
\[ 1_E(\omega)= \begin{cases}1, & \text { if } \omega \in E \\ 0, & \text { if } \omega \notin E\end{cases} \] It's a binary representation of events: - \(1_E(\omega)=1\) : Indicates that the outcome \(\omega\) is part of the event \(E\), meaning \(E\) has occurred. - \(1_E(\omega)=0\) : Indicates that \(\omega\) is not part of \(E\), meaning \(E\) has not occurred.
We know that an event \(E\) is a set of outcomes, but with indicator funciton, we can refer it with the numerical statement that \(1_E(\omega)=\) 1 or 1.
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.
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 \(\sum_{\omega \in \Omega} p(\omega) = 1\).
\(2^\Omega\)是 \(\Omega\) 的幂集(Power Set).↩︎