Sources:

1. 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 $$
$F_X(x)$	Cumulative distribution function (CDF)	Probability that $X$ takes a value less than or equal to $x$, $F_X(x)=(Xx)$
$f_X(x)$	Probability density function (PDF)	Describes the density of $X$ if $X$ is absolutely continuous
$p_X(x)$	Probability mass function (PMF)	Describes the probability of $X$ taking a specific value $x$ if $X$ is discrete
$$	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 $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ where:

• $\mathrm{\Omega }$ is a set (sometimes called a sample space in elementary probability). Elements of $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$. Elements of $\mathcal{F}$ are called events.

• $\mathbb{P}$, the probility measure, is a function from $\mathcal{F}$ to $[0,1]$ with $\mathbb{P}(\mathrm{\Omega })=1$ and such that if events $E_1,E_2,\dots \in \mathcal{F}$ are disjoint, $$\mathbb{P}\left[\bigcup _{j=1}^{\mathrm{\infty }}{E}_j\right]=\sum _{j=1}^{\mathrm{\infty }}\mathbb{P}\left[E_j\right]$$ 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 $\mathrm{\Omega }$ belongs to an event $E$. It is formally defined as:

$$1_E(\omega )=\{\begin{array}{ll}1,& \text{ if }\omega \in E\\ 0,& \text{ if }\omega \notin E\end{array}$$ 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, $\mathrm{\Omega }=\{1,2,3,4,5,6\}$, 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$1_{\mathrm{骰}\mathrm{子}\mathrm{的}\mathrm{结}\mathrm{果}\mathrm{是}\mathrm{奇}\mathrm{数}}(\omega )=1$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 $\mathrm{\Omega }$ is finite or countably infinite. In this case we usually choose $\mathcal{F}$ to be all the subsets of $\mathrm{\Omega }$ (this can be written $\mathcal{F}=2^{\mathrm{\Omega }}$¹)

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

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1. $2^{\mathrm{\Omega }}$是 $\mathrm{\Omega }$ 的幂集(Power Set).↩︎